cartan.math.umb.edu - User contributions [en]

Math 361, Spring 2022

2022-05-12T14:37:08Z

Steven.Jackson:

==Course information==

* See the [http://cartan.math.umb.edu/classes/s22_ma361/s22_ma361_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Mondays, Wednesdays, and Fridays, 11:00 a.m.-11:50 a.m., in W-1-64.
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: W-3-154-27
* Office hours: Mondays, Wednesdays, and Fridays, 12:00 p.m.-12:50 p.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].
* Telephone: (617) 287-6469.

==Important dates==

* Weekly quizzes happen on Wednesdays during the last ten minutes of class. The first quiz is on Wednesday, February 2.
* First midterm: Wednesday, March 2.
* Second midterm: Wednesday, April 13.
* Final exam: Monday, May 16, 11:30 a.m. - 2:30 p.m.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Scoring rubric==

Quizzes and exam questions are all scored on a five-point scale, defined as
follows:

; 5/5 : Response demonstrates substantial mastery of the ideas assessed by the question. May contain small imperfections addressed in comments. Student should move forward and learn new things.
; 4/5 : Response demonstrates understanding of sound technique, but execution errors lead to wrong answer.
; 3/5 : Response is generally on the right track; student would probably solve the problem given sufficient time, but is not yet demonstrating full understanding of the ideas assessed by the problem. Student should spend more time in order to achieve full understanding.
; 2/5 : Response indicates a substantial misconception. Student is unlikely to make progress without first correcting the misconception, and should speak with some other person in order to get back on track.
; 1/5 : Response employs relevant words and phrases, but does not demonstrate a sound understanding of the question or productive approaches to it. Student should seek assistance.
; 0/5 : No response, or response not relevant to the question.

==Weekly assignments==

* [[Math 361, Spring 2022, Assignment 1|Assignment 1]], due Wednesday, February 2.
* [[Math 361, Spring 2022, Assignment 2|Assignment 2]], due Wednesday, February 9.
* [[Math 361, Spring 2022, Assignment 3|Assignment 3]], due Wednesday, February 16.
* [[Math 361, Spring 2022, Assignment 4|Assignment 4]], due Wednesday, February 23.
* [[Math 361, Spring 2022, Assignment 5|Assignment 5]], due Wednesday, March 2.
* [[Math 361, Spring 2022, Assignment 6|Assignment 6]], due Wednesday, March 9.
* [[Math 361, Spring 2022, Assignment 7|Assignment 7]], due Wednesday, March 23.
* [[Math 361, Spring 2022, Assignment 8|Assignment 8]], due Wednesday, March 30.
* [[Math 361, Spring 2022, Assignment 9|Assignment 9]], due Wednesday, April 6.
* [[Math 361, Spring 2022, Assignment 10|Assignment 10]], due Wednesday, April 13.
* [[Math 361, Spring 2022, Assignment 11|Assignment 11]], due Wednesday, April 20.
* [[Math 361, Spring 2022, Assignment 12|Assignment 12]], due Wednesday, April 27.
* [[Math 361, Spring 2022, Assignment 13|Assignment 13]], due Wednesday, May 4.
* [[Math 361, Spring 2022, Assignment 14|Assignment 14]], due Wednesday, May 11.
* [[Math 361, Spring 2022, Assignment 15|Assignment 15]], due before final exam.

==Notes from the final two lectures==

* [http://cartan.math.umb.edu/classes/s22_ma361/notes_splitting.pdf May 9 (on splitting fields)]
* [http://cartan.math.umb.edu/classes/s22_ma361/notes_galois.pdf May 11 (on extension automorphisms and the Galois group)]

Math 361, Spring 2022, Assignment 12

2022-05-12T14:29:25Z

Steven.Jackson:

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Principal ideal domain (a.k.a. ''PID'').

==Carefully state the following theorems (you do not need to prove them):==

# List of units of $F[x]$.
# Theorem relating maximal ideals to irreducible elements (in PIDs).
# Criterion for $F[x]/\left\langle m\right\rangle$ to be a field.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 14

2022-05-12T14:17:49Z

Steven.Jackson: Created page with "__NOTOC__ ==Describe the following procedures:== # Sieve of Eratosthenes (for integers). # Sieve of Eratosthenes (for polynomials with coefficients in a finite field). # Pro..."

__NOTOC__

==Describe the following procedures:==

# Sieve of Eratosthenes (for integers).
# Sieve of Eratosthenes (for polynomials with coefficients in a finite field).
# Procedure to factor polynomials over $\mathbb{C}$.
# Procedure to factor polynomials over $\mathbb{R}$.

==Solve the following problems:==

# Skim the introduction to [https://en.wikipedia.org/wiki/Factorization_of_polynomials the Wikipedia article on polynomial factorization] so you will know where to find search terms when you one day need to know how to factor high-degree polynomials.
# Working over $\mathbb{Z}_2$, factor the polynomial $x^3+1$ into irreducibles. ''(Hint: first look for roots and pull out the corresponding linear factors by long division.)''
# Repeat the previous exercise for $x^4+1$ and for $x^5+1$. ''(Hint: the hardest part will be deciding whether $x^4+x^3+x^2+x+1$ can be factored as the product of two quadratics. But for this, you can make a list of all irreducible quadratics and test for divisibility by each in turn.)''
# Working over $\mathbb{Z}_3$, find all irreducible polynomials of degree two. ''(Hint: you do not need the Sieve; you just need to find quadratics that have no roots.)''
# Construct a field with nine elements.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 15

2022-05-12T13:36:32Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully define the following terms, then give one example and one non-example of each:== # Splitting field (of a non-constant polynomial $f\in F[x]$). # Isomor..."

__NOTOC__

==Carefully define the following terms, then give one example and one non-example of each:==

# Splitting field (of a non-constant polynomial $f\in F[x]$).
# Isomorphism (of field extensions).
# Automorphism (of a field extension).
# $\mathrm{Gal}(F,E,\iota)$ (the ''Galois group'' of the extension $(F,E,\iota)$).
# $\phi(H)$ (the ''fixed field'' of the subgroup $H\leq\mathrm{Gal}(F,E,\iota)$).
# The Galois Correspondence.

==Carefully state the following theorems (you do not need to prove them):==

# Theorem on existence and uniqueness of splitting fields.
# Fundamental Theorem of Galois Theory (this is not actually stated in the notes, but you will find a "summary" of the theorem with certain hypotheses left unstated).

==Solve the following problems:==

# Using the Sieve of Eratosthenes (or any other suitable method, such as root-searching), show that the polynomial $x^3+x+1$ is irreducible over $\mathbb{Z}_2$.
# Show that the quotient ring $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$ is a field. (It is usually denoted $GF(8)$.)
# How many elements does $GF(8)$ have?
# List the elements of $GF(8)$ explicitly.
# Define a function $\phi:GF(8)\rightarrow GF(8)$ by the formula $\phi(x)=x^2$. Show that $\phi$ is a unital ring homomorphism. ''(Hint: to prove that it preserves addition, use the Freshman's Dream.)''
# Compute $\mathrm{ker}(\phi)$.
# Show that $\phi$ is bijective, and hence an isomorphism from $GF(8)$ to itself. (It is usually called the ''Frobenius automorphism''.)
# Make a table of values for $\phi$. ''(This is not as tedious as it appears at first. Remember the Freshman's Dream!)''
# Now define $\iota:\mathbb{Z}_2\rightarrow GF(8)$ by the usual formula $\iota(a)=a+0\alpha+0\alpha^2$, so that $(\mathbb{Z}_2,GF(8),\iota)$ is a field extension. Show that $\phi$ is an automorphism of this extension.
# It is possible to show that $\phi$ generates the whole of $\mathrm{Gal}(\mathbb{Z}_2,GF(8),\iota)$. Taking this for granted, make a group table for this Galois group.
# Find all subgroups of $\mathrm{Gal}(\mathbb{Z}_2,GF(8),\iota)$. ''(Hint: there are very few. Use Lagrange's Theorem!)''
# Compute the Galois Correspondence for $(\mathbb{Z}_2,GF(8),\iota)$.
# (Optional challenge) Repeat the above exercises for $GF(16)$. (That is, first use the Sieve to identify an irreducible quartic in $\mathbb{Z}_2[x]$, then use this quartic to construct a field with sixteen elements, then make tables for the Frobenius automorphism and its powers, and finally compute the Galois Correspondence. This is no more conceptually challenging than for $GF(8)$, but it is somewhat more tedious. However, $(\mathbb{Z}_2,GF(16),\iota)$ is the smallest field extension for which the Galois group has a non-trivial proper subgroup, so it may be of special interest. Though tedious, this example reveals a number of interesting phenomena.)

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022

2022-05-03T22:24:34Z

Steven.Jackson:

Math 361, Spring 2022, Assignment 13

2022-05-01T11:32:25Z

Steven.Jackson:

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Prime ideal.
# Unique factorization domain.
# Principal ideal domain.
# Prime element.
# Proper divisor chain.
# Divisor chain condition.

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating prime ideals to integral domains.
# Theorem relating irreducible elements to maximal ideals ("If $D$ is a principal ideal domain, then $\left\langle m\right\rangle$ is maximal if and only if $m$ is...").
# Theorem relating prime elements to irreducible elements in general.
# Theorem relating prime elements to irreducible elements in principal ideal domains.
# Criteria for $D$ to have unique factorization.
# Classification of ideals in $\mathbb{Z}$ ("$\mathbb{Z}$ is a...").
# Theorem concerning divisor chains in $\mathbb{Z}$ ("$\mathbb{Z}$ has no...").
# Theorem concerning unique factorization in $\mathbb{Z}$.
# Classification of ideals in $F[x]$ ("For any field $F$, $F[x]$ is a...").
# Theorem concerning divisor chains in $F[x]$ ("For any field $F$, $F[x]$ has no...").
# Theorem concerning unique factorization in $F[x]$.

==Solve the following problems:==

# The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{R}[x]$. Find this factorization.
# The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{C}[x]$. Find this factorization.
# '''(The domain $\mathbb{Z}[\sqrt{-5}]$)''' Recall that $\mathbb{Z}[\sqrt{-5}]=\{a+bi\,|\,a,b\in\mathbb{Z}\}$. Show that this set is a unital subring of $\mathbb{C}$, and hence an integral domain.
# Define a function $N:\mathbb{Z}[\sqrt{-5}]\rightarrow\mathbb{Z}_{\geq0}$ by the formula $N(z)=\left\lvert z\right\rvert^2$. (Here the absolute value is taken in the sense of complex numbers, i.e. $\left\lvert a+bi\right\rvert=\sqrt{a^2+b^2}$.) Show that $N$ preserves multiplication, i.e. that $N(z_1z_2)=N(z_1)N(z_2)$.
# Find all elements $a+bi\sqrt{5}\in\mathbb{Z}[\sqrt{-5}]$ with $N(a+bi\sqrt{5})=1$.
# Show that an element of $\mathbb{Z}[\sqrt{-5}]$ is a unit if and only if it has norm one.
# Show that in the ring $\mathbb{Z}[\sqrt{-5}]$, the factorization $a=bc$ is non-trivial if and only if $N(b)<N(a)$ and $N(c)<N(a)$.
# Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $2$. ''(Hint: $N(a+bi\sqrt{5})=a^2+5b^2$, and both $a$ and $b$ are integers.)''
# Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $3$.
# Calculate the norms of the elements $2, 3, 1+i\sqrt{5},$ and $1-i\sqrt{5}$.
# Show that all four of the elements referenced in the previous problem are irreducible in $\mathbb{Z}[\sqrt{-5}]$.
# Show that none of the elements referenced above is prime.
# Show that $\mathbb{Z}[\sqrt{-5}]$ does ''not'' have unique factorization.
# Show that $\mathbb{Z}[\sqrt{-5}]$ ''does'' satisfy the divisor chain condition. ''(Hint: think about norms in a proper divisor chain.)''
# Show that $\mathbb{Z}[\sqrt{-5}]$ must contain at least one non-principal ideal.
# Consider the ideal $J=\left\langle 2,1+i\sqrt{5}\right\rangle=\{2(a+bi\sqrt{5})+(1+i\sqrt{5})(c+di\sqrt{5})\,|\,a,b,c,d\in\mathbb{Z}\}=\{(2a+c-5d)+(2b+c+d)i\sqrt{5}\,|\,a,b,c,d\in\mathbb{Z}\}$. Show that $2\in J$ and $1+i\sqrt{5}\in J$ but $1\not\in J$. ''(Hint: to show that $1\not\in J$, work with the last-given description of the elements of $J$. In order for the coefficient of $i\sqrt{5}$ to vanish, $c$ and $d$ must both be even or both odd. In either case, what is the parity of $2a+c-5d$?)''
# Show that the ideal $J$ is ''not'' principal. ''(Hint: if it were principal, say $J=\left\langle g\right\rangle$, then the generator $g$ would need to be a common divisor of $2$ and $1+i\sqrt{5}$. But these are irreducibles and are not associates of one another. So what are their common divisors?)''
# '''(Optional; a domain with an infinite divisor chain)''' This and all following exercises require some knowledge of complex analysis and are thus optional. In these exercises, if you choose to attempt them, you will construct an example of an infinite proper divisor chain. To begin with, let $R$ denote the set of functions from $\mathbb{C}$ to $\mathbb{C}$ which are complex-analytic at every point. Using the properties of complex derivatives, show that $R$ is a unital ring under pointwise addition and multiplication.
# Show that a non-constant element of $R$ can vanish at only countably many points. ''(Hint: this is the hardest exercise of the whole series. You will need to use the [https://en.wikipedia.org/wiki/Identity_theorem identity theorem] together with the fact that $\mathbb{C}$ is a second-countable topological space.)''
# Show that $R$ is an integral domain. ''(Hint: if $fg=0$ then either $f$ or $g$ must vanish at uncountably many points.)''
# Show that a unit of $R$ cannot have any zeros. ''(Hint: if $f$ has a zero of order $d$ at $z=z_0$, then $1/f$ has a pole of order $d$ at $z=z_0$.)''
# Define a function $f_n:\mathbb{C}\rightarrow\mathbb{C}$ by the formula $f(z)=\sin(z)/\prod_{k=1}^n\left(z-k\pi\right)$. Show that $f_n$ has only removable singularities and thus has a unique extension to an element of $R$ (which we shall also denote by $f_n$).
# Describe the zeros of $f_n$.
# Show that $f_n/f_{n+1}$ has only removable singularities, and thus $f_{n+1}\,|\,f_n$.
# Show that $f_{n+1}\not\sim f_n$. ''(Hint: use the principle, which you proved above, that a unit of $R$ cannot have any zeros.)''
# Conclude that $(f_1,f_2,f_3,\dots)$ is an infinite proper divisor chain in $R$.
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 13

2022-04-30T17:31:08Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully define the following terms, and give one example and one non-example of each:== # Prime ideal. # Unique factorization domain. # Principal ideal domain...."

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Prime ideal.
# Unique factorization domain.
# Principal ideal domain.
# Prime element.
# Proper divisor chain.
# Divisor chain condition.

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating prime ideals to integral domains.
# Theorem relating irreducible elements to maximal ideals ("If $D$ is a principal ideal domain, then $\left\langle m\right\rangle$ is maximal if and only if $m$ is...").
# Theorem relating prime elements to irreducible elements in general.
# Theorem relating prime elements to irreducible elements in principal ideal domains.
# Criteria for $D$ to have unique factorization.
# Classification of ideals in $\mathbb{Z}$ ("$\mathbb{Z}$ is a...").
# Theorem concerning divisor chains in $\mathbb{Z}$ ("$\mathbb{Z}$ has no...").
# Theorem concerning unique factorization in $\mathbb{Z}$.
# Classification of ideals in $F[x]$ ("For any field $F$, $F[x]$ is a...").
# Theorem concerning divisor chains in $F[x]$ ("For any field $F$, $F[x]$ has no...").
# Theorem concerning unique factorization in $F[x]$.

==Solve the following problems:==

# The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{R}[x]$. Find this factorization.
# The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{C}[x]$. Find this factorization.
# '''(The domain $\mathbb{Z}[\sqrt{-5}]$)''' Recall that $\mathbb{Z}[\sqrt{-5}]=\{a+bi\,|\,a,b\in\mathbb{Z}\}$. Show that this set is a unital subring of $\mathbb{C}$, and hence an integral domain.
# Define a function $N:\mathbb{Z}[\sqrt{-5}]\rightarrow\mathbb{Z}_{\geq0}$ by the formula $N(z)=\left\lvert z\right\rvert^2$. (Here the absolute value is taken in the sense of complex numbers, i.e. $\left\lvert a+bi\right\rvert=\sqrt{a^2+b^2}$.) Show that $N$ preserves multiplication, i.e. that $N(z_1z_2)=N(z_1)N(z_2)$.
# Find all elements $a+bi\sqrt{5}\in\mathbb{Z}[\sqrt{-5}]$ with $N(a+bi\sqrt{5})=1$.
# Show that an element of $\mathbb{Z}[\sqrt{-5}]$ is a unit if and only if it has norm one.
# Show that in the ring $\mathbb{Z}[\sqrt{-5}]$, the factorization $a=bc$ is non-trivial if and only if $N(b)<N(a)$ and $N(c)<N(a)$.
# Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $2$. ''(Hint: $N(a+bi\sqrt{5})=a^2+5b^2$, and both $a$ and $b$ are integers.)''
# Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $3$.
# Calculate the norms of the elements $2, 3, 1+i\sqrt{5},$ and $1-i\sqrt{5}$.
# Show that all four of the elements referenced in the previous problem are irreducible in $\mathbb{Z}[\sqrt{-5}]$.
# Show that none of the elements referenced above is prime.
# Show that $\mathbb{Z}[\sqrt{-5}]$ does ''not'' have unique factorization.
# Show that $\mathbb{Z}[\sqrt{-5}]$ ''does'' satisfy the divisor chain condition. ''(Hint: think about norms in a proper divisor chain.)''
# Show that $\mathbb{Z}[\sqrt{-5}]$ must contain at least one non-principal ideal.
# Consider the ideal $J=\left\langle 2,1+i\sqrt{5}\right\rangle=\{2(a+bi\sqrt{5})+(1+i\sqrt{5})(c+di\sqrt{5})\,|\,a,b,c,d\in\mathbb{Z}\}=\{(2a+c-5d)+(2b+c+d)i\sqrt{5}\,|\,a,b,c,d\in\mathbb{Z}\}$. Show that $2\in J$ and $1+i\sqrt{5}\in J$ but $1\not\in J$. ''(Hint: to show that $1\not\in J$, work with the last-given description of the elements of $J$. In order for the coefficient of $i\sqrt{5}$ to vanish, $c$ and $d$ must both be even or both odd. In either case, what is the parity of $2a+c-5d$?)''
# Show that the ideal $J$ is ''not'' principal. ''(Hint: if it were principal, say $J=\left\langle g\right\rangle$, then the generator $g$ would need to be a common divisor of $2$ and $1+i\sqrt{5}$. But these are irreducibles and are not associates of one another. So what are their common divisors?)''
# '''(Optional; a domain with an infinite divisor chain)''' This and all following exercises require some knowledge of complex analysis and are thus optional. In these exercises, if you choose to attempt them, you will construct an example of an infinite proper divisor chain. To begin with, let $R$ denote the set of functions from $\mathbb{C}$ to $\mathbb{C}$ which are complex-analytic at every point. Using the properties of complex derivatives, show that $R$ is a unital ring under pointwise addition and multiplication.
# Show that a non-constant element of $R$ can vanish at only countably many points. ''(Hint: this is the hardest exercise of the whole series. You will need to use the [https://en.wikipedia.org/wiki/Identity_theorem identity theorem] together with the fact that $\mathbb{C}$ is a second-countable topological space.)''
# Show that $R$ is an integral domain. ''(Hint: if $fg=0$ then either $f$ or $g$ must vanish at uncountably many points.)''
# Show that a unit of $R$ cannot have any zeros. ''(Hint: if $f$ has a zero of order $d$ at $z=z_0$, then $1/f$ has a pole of order $d$ at $z=z_0$.)''
# Define a function $f_n:\mathbb{C}\rightarrow\mathbb{C}$ by the formula $f(z)=\sin(z)/\prod_{k=1}^n\left(z-ki\pi\right)$. Show that $f_n$ has only removable singularities and thus has a unique extension to an element of $R$ (which we shall also denote by $f_n$).
# Describe the zeros of $f_n$.
# Show that $f_n/f_{n+1}$ has only removable singularities, and thus $f_{n+1}\,|\,f_n$.
# Show that $f_{n+1}\not\sim f_n$. ''(Hint: use the principle, which you proved above, that a unit of $R$ cannot have any zeros.)''
# Conclude that $(f_1,f_2,f_3,\dots)$ is an infinite proper divisor chain in $R$.
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 11

2022-04-16T20:01:14Z

Steven.Jackson:

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Maximal ideal.
# Divisibility relation (in a domain $D$; i.e. $a|b$ if and only if...).
# Associate relation (in a domain $D$; i.e. $a\sim b$ if and only if...).
# Associate class (of an element of a domain $D$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem characterizing the ideals $I$ for which $R/I$ is a field.
# Containment criterion for principal ideals (i.e. $\left\langle a\right\rangle\subseteq\left\langle b\right\rangle$ if and only if...).
# Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).
# Properties of the associate relation (i.e. $\sim $ is an...).
# Characterization of the associate class $[a]_\sim$
# List of units in $\mathbb{Z}$.

==Solve the following problems:==

# Suppose $u\in F[x]$ is a unit. Prove the $\mathrm{deg}(u)=0$. ''(Hint: start with the equation $u\cdot u^{-1}=1$, and take degrees of both sides.)''
# Suppose $f\in F[x]$ has degree zero. Show that $f$ is a unit. ''(Hint: remember that $F$ is a field. What sort of polynomials have degree zero?)''
# Let $D$ be any integral domain. Prove that $a\in D$ is a unit if and only if $a\sim 1$.
# Suppose $u\in D$ is a unit and $u\sim v$. Prove that $v$ is also a unit.
# An element $a\in D$ is said to be ''irreducible'' if it is not zero, not a unit, and given any factorization $a=bc$, either $b$ is a unit or $c$ is a unit. Describe the irreducible elements of $\mathbb{Z}$.
# Working in $F[x]$ where $F$ is some field, show that any polynomial of degree one is irreducible. ''(Hint: suppose $\deg(f)=1$ and $f=gh$. Taking the degree of both sides of this equation gives $1=\deg(g)+\deg(h)$. What are all the possible values for the ordered pair $(\deg(g),\deg(h))$?)''
# Working in $F[x]$, show that a polynomial $f$ of degree two is irreducible if and only if it has no roots in $F$. ''(Hint 1: you will need the Factor Theorem that you proved in [[Math_361,_Spring_2022,_Assignment_9|Assignment 9]]. Hint 2: suppose you have a factorization $f=gh$ in which neither $g$ nor $h$ is a unit. What are the degrees of $g$ and $h$?)''
# Working in $F[x]$, show that a polynomial $f$ of degree three is irreducible if and only if it has no roots in $F$. ''(Hint: this is very similar to the previous exercise.)''
# Give an example of a field $F$ and a polynomial $f\in F[x]$ of degree four, which has no roots but is nevertheless reducible. ''(Hint: this is much easier than it looks. The most familiar examples are those with $F=\mathbb{R}$. You simply need to find a pair of degree-two polynomials with no roots, and multiply them.)''
# Does the example you produced in the last problem invalidate the reasoning you used in the previous two? If not, at exactly what point does the reasoning you used in the previous two exercises break down in the case of degree-four polynomials?
# Working once more in a general integral domain $D$, prove that if $a$ is irreducible and $a\sim b$, then $b$ is also irreducible.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 11

2022-04-16T20:00:28Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully define the following terms, and give one example and one non-example of each:== # Maximal ideal. # Divisibility relation (in a domain $D$; i.e. $a|b$ i..."

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Maximal ideal.
# Divisibility relation (in a domain $D$; i.e. $a|b$ if and only if...).
# Associate relation (in a domain $D$; i.e. $a\sim b$ if and only if...).
# Associate class (of an element of a domain $D$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem characterizing the ideals $I$ for which $R/I$ is a field.
# Containment criterion for principal ideals (i.e. $\left\langle a\right\rangle\subseteq\left\langle b\right\rangle$ if and only if...).
# Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).
# Properties of the associate relation (i.e. $\sim $ is an...).
# Characterization of the associate class $[a]_\tilde$
# List of units in $\mathbb{Z}$.

==Solve the following problems:==

# Suppose $u\in F[x]$ is a unit. Prove the $\mathrm{deg}(u)=0$. ''(Hint: start with the equation $u\cdot u^{-1}=1$, and take degrees of both sides.)''
# Suppose $f\in F[x]$ has degree zero. Show that $f$ is a unit. ''(Hint: remember that $F$ is a field. What sort of polynomials have degree zero?)''
# Let $D$ be any integral domain. Prove that $a\in D$ is a unit if and only if $a\sim 1$.
# Suppose $u\in D$ is a unit and $u\sim v$. Prove that $v$ is also a unit.
# An element $a\in D$ is said to be ''irreducible'' if it is not zero, not a unit, and given any factorization $a=bc$, either $b$ is a unit or $c$ is a unit. Describe the irreducible elements of $\mathbb{Z}$.
# Working in $F[x]$ where $F$ is some field, show that any polynomial of degree one is irreducible. ''(Hint: suppose $\deg(f)=1$ and $f=gh$. Taking the degree of both sides of this equation gives $1=\deg(g)+\deg(h)$. What are all the possible values for the ordered pair $(\deg(g),\deg(h))$?)''
# Working in $F[x]$, show that a polynomial $f$ of degree two is irreducible if and only if it has no roots in $F$. ''(Hint 1: you will need the Factor Theorem that you proved in [[Math_361,_Spring_2022,_Assignment_9|Assignment 9]]. Hint 2: suppose you have a factorization $f=gh$ in which neither $g$ nor $h$ is a unit. What are the degrees of $g$ and $h$?)''
# Working in $F[x]$, show that a polynomial $f$ of degree three is irreducible if and only if it has no roots in $F$. ''(Hint: this is very similar to the previous exercise.)''
# Give an example of a field $F$ and a polynomial $f\in F[x]$ of degree four, which has no roots but is nevertheless reducible. ''(Hint: this is much easier than it looks. The most familiar examples are those with $F=\mathbb{R}$. You simply need to find a pair of degree-two polynomials with no roots, and multiply them.)''
# Does the example you produced in the last problem invalidate the reasoning you used in the previous two? If not, at exactly what point does the reasoning you used in the previous two exercises break down in the case of degree-four polynomials?
# Working once more in a general integral domain $D$, prove that if $a$ is irreducible and $a\sim b$, then $b$ is also irreducible.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 10

2022-04-09T12:13:19Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully define the following terms, and give one example and one non-example of each:== # Standard representative (of an element of $F[x]/\left\langle m\right\..."

__NOTOC__

==Carefully define the following terms, and give one example and one non-example of each:==

# Standard representative (of an element of $F[x]/\left\langle m\right\rangle$; i.e. the representative whose uniqueness is guaranteed by the theorem concerning unique representation below).
# Standard generator (of $F[x]/\left\langle m\right\rangle$; usually this is denoted by $\alpha$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning unique representation of elements of $F[x]/\left\langle m\right\rangle$.
# Theorem concerning $m(\alpha)$ (where $\alpha$ is the standard generator of $F[x]/\left\langle m\right\rangle$).

==Carefully describe the following procedures:==

# Procedure to calculate the standard representation of the product $(f+\left\langle m\right\rangle)(g+\left\langle m\right\rangle)$ (i.e. the "machine implementation" of multiplication in $F[x]/\left\langle m\right\rangle$).
# Procedure to rewrite "high" powers of the standard generator $\alpha$ in terms of lower powers, using the theorem concerning $m(\alpha)$ (i.e. the "human implementation" of multiplication in $F[x]/\left\langle m\right\rangle)$).

==Solve the following problems:==

# Let $R$ denote the quotient ring $\mathbb{Z}_2[x]/\left\langle x^2+1\right\rangle$. List the elements of $R$, then make a multiplication table. Is $R$ a field?
# Let $GF(8)$ denote the quotient ring $\mathbb{Z}_2[x]/\left\langle x^3+x+1\right\rangle$. List the elements of $GF(8)$. Be sure to list each element only once. (You will probably find it more pleasant to write them in terms of the standard generator $\alpha$ rather than using coset notation.)
# Working in $GF(8)$, compute the sum $(1+\alpha^2)+(1+\alpha)$.
# Using the "machine implementation" of multiplication in $GF(8)$, compute the product $(1+\alpha^2)(1+\alpha)$. Be sure to write your answer in its standard representation.
# Working in $GF(8)$, find a formula for $\alpha^3$ in terms of lower powers of $\alpha$. ''(Hint: use the theorem regarding $m(\alpha)$.)''
# Use the formula you found above to compute the standard representations of $\alpha^4, \alpha^5,$ $\alpha^6,$ and $\alpha^7$.
# Redo your calculation of $(1+\alpha^2)(1+\alpha)$, this time avoiding the "machine implementation" in favor of the formula you found above for $\alpha^3$. Verify that you obtain the same answer.
# Suppose $m\in\mathbb{Z}_p[x]$ is a polynomial of degree $d$. Compute the cardinality of the ring $\mathbb{Z}_p[x]/\left\langle m\right\rangle$. ''(Hint: use the theorem on unique representation of elements. How many choices are there for each coefficient, and how many coefficients are there?)''
# Verify that the formula you found above correctly predicts the number of elements of $GF(8)$.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 9

2022-04-02T11:43:37Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 23, first two pages (on the division algorithm). ==Carefully define the following terms, and give one example and one non-example of each:==..."

__NOTOC__

==Read:==

# Section 23, first two pages (on the division algorithm).

==Carefully define the following terms, and give one example and one non-example of each:==

# Degree (of a polynomial; please be sure to include the case of the zero polynomial).
# Constant polynomial.
# Divisibility relation on polynomials.
# $f\,\%\,g$.

==Carefully state the following theorems (you do not need to prove them):==

# Degree bounds on sum and product (general form).
# Formula for $\mathrm{deg}(fg)$ when $R$ is an integral domain.
# Theorem concerning zero-divisors in $D[x]$ when $D$ is an integral domain (i.e. "If $D$ is an integral domain then so is...")
# Theorem on polynomial long division.
# Divisibility test for polynomials with coefficients in a field.

==Solve the following problems:==

# Section 23, problems 1, 2, 3, and 4.
# Working in $\mathbb{Q}[x]$, find the remainder when $f(x)=x^2+x-3$ is divided by $x-5$. Then compute $f(5)$.
# Working in $\mathbb{Z}_7[x]$, find the remainder when $f(x)=x^3+4x+1$ is divided by $x-2$. Then compute $f(2)$.
# Using the theorem on polynomial long division, prove the conjecture suggested by the last two exercises.
# Prove the ''Factor Theorem:'' if $F$ is any field, and $f\in F[x]$ is any polynomial with coefficients in $F$, then $f(a)=0$ if and only if $x-a$ is a factor of $f$ (i.e. $f$ is a multiple of $x-a$).
# A ''root'' of a polynomial $f\in F[x]$ is an element $a\in F$ such that $f(a)=0$. Prove that a polynomial of degree $n$ has at most $n$ roots. ''(Hint: begin by assuming that the roots of $f$ are $a_1,\dots,a_r$ and then prove that $\mathrm{deg}(f)\geq r$.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 8

2022-03-27T23:44:52Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 21. # Section 22. ==Carefully define the following terms, and give one example and one non-example of each:== # Formal fraction (from an inte..."

__NOTOC__

==Read:==

# Section 21.
# Section 22.

==Carefully define the following terms, and give one example and one non-example of each:==

# Formal fraction (from an integral domain $D$).
# Equivalence (of formal fractions).
# Fraction (from an integral domain $D$).
# Addition (of fractions).
# Multiplication (of fractions).
# $\mathrm{Frac}(D)$ (the ''field of fractions'' of the integral domain $D$).
# Canonical injection (of an integral domain $D$ into its field of fractions).
# Polynomial function (from a ring $R$ into itself).
# Polynomial expression (with coefficients in a ring $R$).
# Addition (of polynomial expressions).
# Multiplication (of polynomial expressions).
# $R[x]$ (the ''ring of polynomial expressions, with coefficients in $R$, in the indeterminate $x$,'' or "$R$ adjoin $x$" for short).

==Carefully state the following theorems (you do not need to prove them):==

# Equality test for fractions.
# Universal mapping property of $\mathrm{Frac}(D)$.
# Example of two distinct polynomial expressions that give rise to the same polynomial function.

==Solve the following problems:==

# Section 21, problems 1 and 2 ''(translation of problems: you are being asked to describe the concrete model of the field of fractions of the given integral domain $D$ arising from the inclusion of $D$ in the given field $F$).''
# Section 22, problems 1, 2, 3, 4, 5, and 6.
# (Rational expressions). Next week we shall prove that whenever $D$ is an integral domain, so is $D[x]$. For purposes of this exercise, you may take this fact for granted. Thus, the field of fractions of $D[x]$ is a well-defined object, which is usually denoted $D(x)$. Write down two "random" elements of the field $\mathbb{R}(x)$, and show how to add them, and also how to multiply them.
# (An infinite ring with positive characteristic). Let $R=\mathbb{Z}_3[x]$ denote the ring of polynomial expressions with coefficients in $\mathbb{Z}_3$. Write the table of values of the initial morphism $\iota:\mathbb{Z}\rightarrow R$, and show that $\mathrm{char}(R)=3$.
# Let $R$ be as in the previous exercise. Show that $R$ is an infinite ring, even though it has characteristic three and its prime subring is thus a copy of $\mathbb{Z}_3$.
# Let $R$ be as in the previous exercise and put $F=\mathrm{Frac}(R)$. (We will show next week that $R$ is an integral domain; for purposes of this problem you may take this for granted.) Show that $F$ is an infinite field of positive characteristic.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 6

2022-03-14T19:29:37Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully define the following terms, then give one example and one non-example of each:== # Euler totient function (as examples please give one or two illustrat..."

__NOTOC__

==Carefully define the following terms, then give one example and one non-example of each:==

# Euler totient function (as examples please give one or two illustrative calculations of its values; non-examples are not sensible or needed in this case).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem characterizing the units of $\mathbb{Z}_n$ (i.e. $[a]\in\mathbb{Z}_n$ is a unit if and only if...).
# Formula for $\phi(p^k)$ when $p$ is prime.
# Formula for $\phi(ab)$ when $\mathrm{gcd}(a,b)=1$.
# Formula for $\phi(n)$ when the prime factorization $n=p_1^{k_1}\dots p_l^{k_l}$ is known.

==Solve the following problems:==

# Make a table showing the values of $\phi(n)$ for $n\in\{1,2,3,\dots,20\}$.
# Two students try to calculate $\phi(45)$ as follows: one says that $\phi(45)=\phi(9)\phi(5)=(3^2-3^1)(5-1)=6\times4=24$, and another says that $\phi(45)=\phi(3)\phi(15)=\phi(3)\phi(3)\phi(5)=(3-1)(3-1)(5-1)=16$. Which one is wrong, and why?
# The table you constructed above should show that $\phi(15)=8$. Working in $\mathbb{Z}_{15}$, compute the following expressions: $1^8, 2^8, 4^8, 7^8, 8^8, 11^8, 13^8,$ and $14^8$. ''(Hint: there are various tricks that make these computations easier than they look. For example, when computing powers of $4$ you will quickly find that $4^2=16=1$, from which it follows that $4^8=(4^2)^4=1^4=1$. For another example, when computing powers of $14$ it will help to notice that $14=-1$. Using tricks like this, a clever person can compute all of these expressions with very little work.)''
# Based on the previous problem, try to formulate a conjecture regarding the value of the expression $a^{\phi(n)}$ in $\mathbb{Z}_n$.
# Try to prove the conjecture you formulated above. ''(Hint: Lagrange's Theorem is very, very helpful.)''
# Again working in $\mathbb{Z}_{15}$, compute the expressions $3^8, 5^8, 6^8, 9^8, 10^8,$ and $12^8$. Do these contradict the conjecture you formulated above? (If so, then reformulate the conjecture. If your initial conjecture was wrong, then reformulating it may give you a crucial hint about how to prove the reformulated conjecture, since any successful proof will need to make some use of the additional hypothesis. The process of mathematical discovery often works this way---it is good to learn from one's mistakes.)

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 5

2022-03-14T19:28:04Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 18. # Section 19. ==Carefully define the following terms, then give one example and one non-example of each:== # The ''initial morphism'' fro..."

__NOTOC__

==Read:==

# Section 18.
# Section 19.

==Carefully define the following terms, then give one example and one non-example of each:==

# The ''initial morphism'' from $\mathbb{Z}$ to any unital ring $R$.
# $\mathrm{char}(R)$ (the ''characteristic'' of a unital ring $R$).
# The ''prime subring'' of a unital ring $R$.
# Zero-divisor (in a commutative ring $R$).
# Integral domain.
# Field.

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating the prime subring to the characteristic (i.e. "The prime subring of a unital ring $R$ is an isomorphic copy of...")
# Formula for $\mathrm{char}(\mathbb{Z}_a\times\mathbb{Z}_b)$.
# Chinese Remainder Theorem.
# Theorem concerning the characteristic of an integral domain.

==Solve the following problems:==

# Section 18, problems 15, 17, 18, and 40.
# Section 19, problems 1, 2, 5, 7, 9, and 11.
# (The Freshman's Dream) Suppose that $R$ is a commutative, unital ring of characteristic two, and choose any $a,b\in R$. Prove that $(a+b)^2=a^2+b^2$. ''(Please do not reveal this theorem to actual freshmen, who must work in rings of characteristic zero and who already have enough trouble squaring binomials correctly.)''
# (The Freshman's Dream in general) Generalize the above exercise as follows: let $R$ be a commutative, unital ring of prime characteristic $p$, and let $a,b\in R$ be arbitrary. Prove that $(a+b)^p=a^p+b^p$. ''(Hint: use the [https://en.wikipedia.org/wiki/Binomial_theorem binomial theorem], which is valid in any commutative ring.)''
# Give an example to show that the Freshman's Dream does ''not'' hold in composite characteristic.
# Suppose that $R$ is a commutative, unital ring, and that $a\in R$ is a unit. Show that $a$ is ''not'' a zero-divisor. ''(Hint: suppose to the contrary that there exists $b\neq0$ with $ab=0$. What happens if you multiply this equation by $a^{-1}$?)''
# Prove that every field is an integral domain.
# Generalize the above result by showing that any unital subring of a field is an integral domain. ''(Hint: Suppose that $F$ is a field and $R$ is a unital subring of $F$. If $R$ had zero-divisors, then they would also be zero-divisors in $F$.)''
# Suppose that $D$ is an integral domain. Show that $\mathrm{char}(D)$ is either zero or a prime. ''(Hint: suppose to the contrary that $\mathrm{char}(D)$ is composite, say $\mathrm{char}(D)=nm$ for $n,m>1$ and let $\iota$ be the initial morphism. What is $\iota(n)\cdot\iota(m)$?)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 7

2022-03-14T17:56:43Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 20. ==Carefully define the following terms, and give one example and one non-example of each:== # Private key (in the RSA cryptosystem; i.e. "..."

__NOTOC__

==Read:==

# Section 20.
==Carefully define the following terms, and give one example and one non-example of each:==

# Private key (in the RSA cryptosystem; i.e. "The private key is the ordered pair consisting of...").
# Public key (in the RSA cryptosystem).
# Formal fraction (from an integral domain $D$).
# Equivalence (of formal fractions).
# Fraction (from an integral domain $D$).
# $\mathrm{Frac}(D)$ (the ''field of fractions'' of the integral domain $D$).

==Carefully state the following theorems (you do not need to prove them):==

# Euler's Theorem.
# Equality test for fractions.

==Solve the following problems:==

# Section 20, problems 5 and 10.
# Taking $p=5$ and $q=7$, generate a public/private keypair for the RSA cryptosystem. (Hint: start by choosing an encryption exponent $e$ which is relatively prime to $\phi(35)=24$. If you have trouble generating the corresponding decryption exponent, use $e=5$, which is easy to invert modulo $24$ by inspection.)
# Using the public key generated above, encrypt the "message" $m=3$.
# Using the private key generated above, decrypt the "ciphertext" $c$ generated by the previous problem.
# You have undoubtedly noticed that your public and private keys are identical, which is undesirable in an allegedly asymmetric cryptosystem. In fact, for these particular choices of $p$ and $q$, the two keys will be identical regardless of which encryption exponent is chosen. Try to explain why. ''(Hint: investigate the structure of the group $\mathcal{U}(\mathbb{Z}_{24})$ using the Chinese Remainder Theorem.)''
# Repeat the previous exercises with slightly larger choices of $p$ and $q$ until you find a keypair in which the keys are distinct. (You may wish to use a machine to help with the arithmetic.)
# Let $D$ denote the ring of real-valued polynomial functions. (We will see next week that this is an integral domain; for purposes of this problem you may take that fact for granted.) Write down some fractions from $D$. What did you call objects of this type when you were in high school?
# With $D$ as above, prove that the fractions $\frac{x^2-1}{x^2-2x+1}$ and $\frac{x+1}{x-1}$ are equal.
# Suppose that $D$ is any integral domain and that $a,b,c\in D$ with $a\neq0$ and $b\neq0$. Prove the '''cancellation property of fractions,''' that $\frac{ab}{ac}=\frac{b}{c}$.
# Suppose that $D$ is any integral domain and that $a,b,c\in D$ with $b\neq0$ and $a$ a unit. Prove that $\frac{ab}{c}=\frac{b}{a^{-1}c}$ and that $\frac{b}{ac}=\frac{a^{-1}b}{c}$.
# '''(Addition with a common denominator)''' Starting from the definition of addition in $\mathrm{Frac}(D)$, show that $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 4

2022-02-21T21:39:07Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 26. ''(Note that we are skipping forward somewhat in the text; however we will shortly return to the earlier material in Section 20.)'' ==Car..."

__NOTOC__

==Read:==

# Section 26. ''(Note that we are skipping forward somewhat in the text; however we will shortly return to the earlier material in Section 20.)''

==Carefully define the following terms, then give one example and one non-example of each:==

# Homomorphism (of rings).
# Unital homomorphism (of unital rings).
# Pushforward (of a subring under a homomorphism; a.k.a. ''forward image'').
# Pullback (of a subring under a homomorphism; a.k.k. ''pre-image'').
# Image (of a ring homomorphism).
# Kernel (of a ring homomorphism).
# Ideal.
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
# Addition (in $R/I$, i.e. coset addition).
# Multiplication (in $R/I$, i.e. coset multiplication).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning $\phi(0_R)$, where $\phi:R\rightarrow S$ is a ring homomorphism.
# Examples of ring homomorphisms $\phi:R\rightarrow S$ to show that $\phi(1_R)$ ''may or may not'' equal $1_S$, even when $R$ and $S$ are both unital.
# Theorem characterizing the properties of the pushforward of a subring (i.e. "The pushforward of a subring is a...").
# Theorem characterizing the properties of the pullback of a subring (i.e. "The pullback of a subring is a...").
# Theorem characterizing the special properties of kernels (i.e. "Kernels absorb..." or "Kernels are...").
# Theorem characterizing ideals which contain units.
# Theorem characterizing the ideals of a field.
# Theorem characterizing when coset multiplication is well-defined (i.e. "Multiplication in $R/I$ is well-defined provided that $I$ is an...").
# Equality test for elements of $R/I$.

==Solve the following problems:==

# Section 26, problems 4, 17, and 18 ''(hint: if $\phi:F\rightarrow S$ is a homomorphism defined on a field $F$, then there are not many possibilities for $\mathrm{ker}(\phi)$)''.
# '''(Canonical projection)''' Suppose $I$ is an ideal of a ring $R$. Define a map $\pi:R\rightarrow R/I$ by the formula $\pi(r)=r+I$. Show that $\pi$ is an epimorphism, and that it is a unital epimorphism whenever $R$ is a unital ring.
# Let $\pi:R\rightarrow R/I$ be the canonical projection defined above. Calculate $\mathrm{ker}(\pi)$.
# Prove that $R/\{0\}$ is always isomorphic to $R$ itself. ''(Hint: use the your calculation of $\mathrm{ker}(\pi)$ from the last problem.)''
# Prove that $R/R$ is always a zero ring. ''(Hint: use the equality test for cosets.)''
# We shall see next week that there is one and only one ring homomorphism $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_3$ for which $\phi(1)=(1,1)$. Write the table of values for this homomorphism, then describe $\mathrm{im}(\phi)$ and $\mathrm{ker}(\phi)$.
# Repeat the above exercise with $\mathbb{Z}_2\times\mathbb{Z}_4$ in place of $\mathbb{Z}_2\times\mathbb{Z}_3$.
# By comparing the previous two exercises, see whether you can make any conjecture about the relationship between $\mathbb{Z}_{a}\times\mathbb{Z}_b$ and $\mathbb{Z}_{ab}$. (If you manage this then you will have re-discovered the ancient and beautiful ''Chinese Remainder Theorem'' (CRT), which we will study next week.)

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 3

2022-02-12T14:30:16Z

Steven.Jackson:

__NOTOC__

==Read:==

# Section 19.

==Carefully define the following terms, then give one example and one non-example of each:==

# Zero-divisor.
# Integral domain.
# Field.
# Subring.
# Zero (a.k.a. ''trivial'') subring.
# Improper subring.
# Subring generated by a subset.
# Prime subring (of a unital ring).

==Carefully state the following theorems (you do not need to prove them):==

# Zero-product property (of integral domains).
# Cancellation law (in integral domains).
# Theorem relating fields to integral domains.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain.

==Solve the following problems:==

# Section 19, problems 1, 2, 3, 4, and 14 ''(hint for 14: to show that this matrix is a left zero-divisor, find another matrix whose image in contained in the kernel of this one; to show that it is a right zero-divisor, find another matrix whose kernel contains the image of this one)''.
# Describe the prime subrings of $\mathbb{Q}$, of $\mathbb{R}$, and of $\mathbb{C}$.
# Describe the prime subring of $\mathbb{Z}$.
# Describe the prime subring of $\mathbb{Z}_n$.
# Working in the field $\mathbb{Z}_3$, solve the equation $x^3=x$.
# Working in the field $\mathbb{Z}_5$, solve the equation $x^5=x$.
# Working in the field $\mathbb{Z}_7$, solve the equation $x^7=x$.
# By now you probably have a conjecture about $\mathbb{Z}_{11}$. Do not try to prove this. Instead, prove the conjecture for $\mathbb{Z}_p$ where $p$ is an arbitrary prime. ''(Hint: the conjecture is obviously true if $x=0$. Otherwise $x$ is an element of the group of units of $\mathbb{Z}_p$ (why?). But as we have seen, Lagrange's Theorem implies that in ''any'' group $G$ we have $g^{\left\lvert G\right\rvert}=e$ for every $g\in G$. This gives rise to a certain identity for non-zero elements of $\mathbb{Z}_p$. Multiplying both sides of this identity by $x$ will prove the conjecture.)''
# Show by a simple counterexample (e.g. in $\mathbb{Z}_6$) that the result above is ''not'' generally true in $\mathbb{Z}_n$ when $n$ is composite. Exactly which part of your proof above breaks in the composite case?
# Try to correctly generalize the conjecture to the composite case (i.e. formulate and prove a statement which encompasses the prime case but is also true in the composite case). In doing this you will be following in the footsteps of Leonhard Euler; this result (like many others) is known as ''Euler's Theorem,'' and it is in fact the mathematical basis of RSA encryption.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 3

2022-02-12T14:26:06Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 19. ==Carefully define the following terms, then give one example and one non-example of each:== # Zero-divisor. # Integral domain. # Field...."

__NOTOC__

==Read:==

# Section 19.

==Carefully define the following terms, then give one example and one non-example of each:==

# Zero-divisor.
# Integral domain.
# Field.
# Subring.
# Zero (a.k.a. ''trivial'') subring.
# Improper subring.
# Subring generated by a subset.
# Prime subring (of a unital ring).

==Carefully state the following theorems (you do not need to prove them):==

# Zero-product property (of integral domains).
# Cancellation law (in integral domains).
# Theorem relating fields to integral domains.
# Theorem characterizing the units and zero-divisors of $mathbb{Z}_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain.

==Solve the following problems:==

# Section 19, problems 1, 2, 3, 4, and 14 ''(hint for 14: to show that this matrix is a left zero-divisor, find another matrix whose image in contained in the kernel of this one; to show that it is a right zero-divisor, find another matrix whose kernel contains the image of this one)''.
# Describe the prime subrings of $\mathbb{Q}$, of $\mathbb{R}$, and of $\mathbb{C}$.
# Describe the prime subring of $\mathbb{Z}$.
# Describe the prime subring of $\mathbb{Z}_n$.
# Working in the field $\mathbb{Z}_3$, solve the equation $x^3=x$.
# Working in the field $\mathbb{Z}_5$, solve the equation $x^5=x$.
# Working in the field $\mathbb{Z}_7$, solve the equation $x^7=x$.
# By now you probably have a conjecture about $\mathbb{Z}_{11}$. Do not try to prove this. Instead, prove the conjecture for $\mathbb{Z}_p$ where $p$ is an arbitrary prime. ''(Hint: the conjecture is obviously true if $x=0$. Otherwise $x$ is an element of the group of units of $\mathbb{Z}_p$ (why?). But as we have seen, Lagrange's Theorem implies that in ''any'' group $G$ we have $g^{\left\lvert G\right\rvert}=e$ for every $g\in G$. This gives rise to a certain identity for non-zero elements of $\mathbb{Z}_p$. Multiplying both sides of this identity by $x$ will prove the conjecture.)''
# Show by a simple counterexample (e.g. in $\mathbb{Z}_6$) that the result above is ''not'' generally true in $\mathbb{Z}_n$ when $n$ is composite. Exactly which part of your proof above breaks in the composite case?
# Try to correctly generalize the conjecture to the composite case (i.e. formulate and prove a statement which encompasses the prime case but is also true in the composite case). In doing this you will be following in the footsteps of Leonhard Euler; this result (like many others) is known as ''Euler's Theorem,'' and it is in fact the mathematical basis of RSA encryption.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 2

2022-02-05T16:24:02Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 18. ==Carefully define the following terms, then give one example and one non-example of each:== # Ring. # Unital (ring). # Commutative (ring..."

__NOTOC__

==Read:==

# Section 18.

==Carefully define the following terms, then give one example and one non-example of each:==

# Ring.
# Unital (ring).
# Commutative (ring).

==Carefully state the following theorems (you do not need to prove them):==

# Rules of sign for rings (this appears in the text as Theorem 18.8).

==Solve the following problems:==

# '''(Direct products of groups)''' Let $G$ and $H$ be groups. Define a binary operation on the Cartesian product set $G\times H$ by the formula $(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1h_2)$. Show that (i) this operation is associative; (ii) the ordered pair $(e_G,e_H)$ is an identity element for this operation; and (iii) any ordered pair $(g,h)$ has inverse $(g^{-1},h^{-1})$. Thus, with this operation, the Cartesian product $G\times H$ becomes a group, which we call the ''direct product'' of the groups $G$ and $H$.
# List the elements of $\mathbb{Z}_2\times\mathbb{Z}_2$, and then make an operation table for the operation defined above. ''(Note: with this we have at last proved that the Klein four-group is really a group, in particular that its operation is really associative.)''
# Is the direct product group $\mathbb{Z}_2\times\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_4$? Prove your answer.
# '''(Direct products of rings)''' Now let $R$ and $S$ be rings. Define two binary operations on the Cartesian product set $R\times S$ by the formulas $(r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)$ and $(r_1,s_1)(r_2,s_2)=(r_1r_2,s_1s_2)$. Show that with these operations, $R\times S$ also becomes a ring, also called the ''direct product'' of $R$ and $S$.
# Show that if $R$ and $S$ are both commutative, then so is $R\times S$.
# Show that if $R$ and $S$ are both unital, then so is $R\times S$.
# Make a multiplication table for the ring $\mathbb{Z}_2\times\mathbb{Z}_2$, and explicitly identify the unity element of this ring.
# '''(Zero-product property fails in direct products)''' The real number system has the well-known ''zero-product property:'' if $xy=0$ then either $x=0$ or $y=0$. Prove that this is ''not'' true in arbitrary rings, by giving an explicit counterexample in the ring $\mathbb{Z}_2\times\mathbb{Z}_2$.
# '''(Zero-product property also fails for functions)''' Now consider the ring $\mathrm{Fun}(\mathbb{R},\mathbb{R})$ with the usual "pointwise" operations that we discussed in class. Show that the zero-product property also fails in this ring, by giving two specific non-zero functions $f$ and $g$ with $fg=0$. ''(Hint: you will almost certainly want $f$ and $g$ to be "piecewise-defined" functions.)''
# '''(The zero ring)''' Suppose $R$ is any set with a single element. Show that there is one and only one way of defining binary operations $+$ and $\cdot$ on $R$ which turn $R$ into a ring. Make operation tables for $+$ and $\cdot$. (Any ring with only a single element is called a ''zero ring.'' Once we have defined what we mean by an "isomorphism" of rings, we will prove that all zero rings are isomorphic with one another, and because of this we will sometimes speak of ''the'' zero ring rather than ''a'' zero ring.)
# '''(The zero ring is unital)''' Show that every zero ring is in fact unital. Prove that in any zero ring, one has the surprising equality $0_R=1_R$.
# '''(The zero ring is the only ring in which $0_R=1_R$)''' Suppose now that $R$ is ''any'' unital ring in which $0_R=1_R$. Prove that $R$ has only one element, and is thus a zero ring. ''(Hint: you will need to use at least one of the three assertions in the Rules of Sign theorem referenced above.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 1

2022-01-29T03:49:43Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 14. ==Carefully define the following terms, then give one example and one non-example of each:== # $H\leq G$ # $H\trianglelefteq G$. # Coset..."

__NOTOC__

==Read:==

# Section 14.

==Carefully define the following terms, then give one example and one non-example of each:==

# $H\leq G$
# $H\trianglelefteq G$.
# Coset multiplication (when $H\trianglelefteq G$).
# Canonical projection (from $G$ onto $G/H$).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem characterizing when coset multiplication is well-defined.
# Theorem concerning the properties of coset multiplication ("When $H\trianglelefteq G$, coset multiplication turns $G/H$ into a...").
# Theorem describing the kernel of the canonical projection.
# Fundamental Theorem of Homomorphisms.

==Solve the following problems:==

# Make an operation table for the quotient group $\mathbb{Z}_{12}/\left\langle 4\right\rangle$.
# Section 14, problems 1, 9, 24, 30 ''(hint: in December we used Lagrange's Theorem to prove that for any finite group $G$ and any $g\in G$, one always has $g^{\left\lvert G\right\rvert}=e$; now apply similar reasoning to the group $G/H$)'', and 31.
# Consider the group $D_4$ of symmetries of a square; for purposes of this problem we will use the notation introduced on page 80 of the text. Let $H$ denote the subgroup $\left\langle\delta_1,\delta_2\right\rangle$ generated by reflections in the diagonals. Determine whether $H$ is a normal subgroup of $D_4$. If it is a normal subgroup, then write the operation table for the quotient group $D_4/H$. (Warning: normality can be checked by brute force, but this is very tedious and there is a shorter way. In the next problem you will actually need the brute-force check but it will be much shorter.)
# Repeat the above exercise for the subgroup $\left\langle\delta_1\right\rangle$ generated by $\delta_1$ alone.
# Let $\pi:\mathbb{Z}_{12}\rightarrow\mathbb{Z}_{12}/\left\langle 4\right\rangle$ denote the canonical projection. Write the table of values for $\pi$.
# Let $G$ be any group. Prove that the trivial subgroup $\{e\}$ is normal in $G$. Then prove that the quotient group $G/\{e\}$ is isomorphic to $G$ itself. ''(Hint: you need to show that the canonical projection is injective in this case.)''
# Let $G$ be any group. Prove that the improper subgroup $G$ is normal in $G$. Then write the operation table for the quotient group $G/G$.
# Recall that $\mathbb{R}^*$ denotes the set of ''non-zero'' real numbers, regarded as a group under ordinary multiplication, and let $\phi:S_n\rightarrow\mathbb{R}^*$ denote the ''sign homomorphism'' that takes even permutations to $1$ and odd permutations to $-1$. Compute $\ker(\phi)$, write an operation table for the quotient group $S_n/\ker(\phi)$, and give a table of values for the momomorphism $\widehat{\phi}:S_n/\ker(\phi)\rightarrow\mathbb{R}^*$ whose existence is asserted by the Fundamental Theorem.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Main Page

2022-01-24T01:33:01Z

Steven.Jackson:

__NOTOC__
==Course Pages==

===Spring 2022===

* [[Math 361, Spring 2022|Math 361]], Abstract Algebra II

===Fall 2021===

* [[Math 360, Fall 2021|Math 360]], Abstract Algebra I

===Spring 2021===

* [[Math 361, Spring 2021|Math 361]], Abstract Algebra II

===Fall 2020===

* [[Math 360, Fall 2020|Math 360]], Abstract Algebra I

===Spring 2020===

* [[Math 361, Spring 2020|Math 361]], Abstract Algebra II

===Fall 2019===

* [[Math 260, Fall 2019|Math 260]], Linear Algebra I
* [[Math 360, Fall 2019|Math 360]], Abstract Algebra I

===Spring 2019===

* [[Math 242, Spring 2019|Math 242]], Multivariable and Vector Calculus
* [[Math 361, Spring 2019|Math 361]], Abstract Algebra II

===Fall 2018===

* [[Math 260, Fall 2018|Math 260]], Linear Algebra I
* [[Math 360, Fall 2018|Math 360]], Abstract Algebra I

===Spring 2018===

* [[Math 361, Spring 2018|Math 361]], Abstract Algebra II
* [[Math 380, Spring 2018|Math 380]], Introduction to Computational Algebraic Geometry

===Fall 2017===

* [[Math 260, Fall 2017|Math 260]], Linear Algebra I
* [[Math 360, Fall 2017|Math 360]], Abstract Algebra I

===Spring 2017===

* [[Math 260, Spring 2017|Math 260]], Linear Algebra I
* [[Math 361, Spring 2017|Math 361]], Abstract Algebra II

===Fall 2016===

* [[Math 360, Fall 2016|Math 360]], Abstract Algebra I
* [[Math 480, Fall 2016|Math 480]], Information Theory

===Spring 2016===

* [[Math 141, Spring 2016|Math 141]], Calculus II
* [[Math 361, Spring 2016|Math 361]], Abstract Algebra II

===Fall 2015===

* [[Math 260, Fall 2015|Math 260]], Linear Algebra I
* [[Math 360, Fall 2015|Math 360]], Abstract Algebra I

===Spring 2015===

* [[Math 361, Spring 2015|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2015|Math 480]], Introduction to Cryptography

===Fall 2014===

* [[Math 360, Fall 2014|Math 360]], Abstract Algebra I
* [[Math 440, Fall 2014|Math 440]], General Topology

===Spring 2014===

* [[Math 361, Spring 2014|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2014|Math 480]], Introduction to Computational Algebraic Geometry II

===Fall 2013===

* [[Math 242, Fall 2013|Math 242]], Calculus III
* [[Math 360, Fall 2013|Math 360]], Abstract Algebra I

===Spring 2013===

* [[Math 361, Spring 2013|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2013|Math 480]], Introduction to Computational Algebraic Geometry

==Helpful Links==

* Summary of the [http://en.wikipedia.org/wiki/Help:Wiki_markup Mediawiki markup language].
* Tips on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

Math 361, Spring 2022

2022-01-24T01:31:11Z

Steven.Jackson: Created page with "==Course information== * See the [http://cartan.math.umb.edu/classes/s22_ma361/s22_ma361_syllabus.pdf syllabus] for general information and the schedule of readings. * Class..."

==Course information==

* See the [http://cartan.math.umb.edu/classes/s22_ma361/s22_ma361_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Mondays, Wednesdays, and Fridays, 11:00 a.m.-11:50 a.m., in W-1-64.
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: W-3-154-27
* Office hours: Mondays, Wednesdays, and Fridays, 12:00 p.m.-12:50 p.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].
* Telephone: (617) 287-6469.

==Important dates==

* Weekly quizzes happen on Wednesdays during the last ten minutes of class. The first quiz is on Wednesday, February 2.
* First midterm: Wednesday, March 2.
* Second midterm: Wednesday, April 13.
* Final exam: To be announced.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Scoring rubric==

Quizzes and exam questions are all scored on a five-point scale, defined as
follows:

; 5/5 : Response demonstrates substantial mastery of the ideas assessed by the question. May contain small imperfections addressed in comments. Student should move forward and learn new things.
; 4/5 : Response demonstrates understanding of sound technique, but execution errors lead to wrong answer.
; 3/5 : Response is generally on the right track; student would probably solve the problem given sufficient time, but is not yet demonstrating full understanding of the ideas assessed by the problem. Student should spend more time in order to achieve full understanding.
; 2/5 : Response indicates a substantial misconception. Student is unlikely to make progress without first correcting the misconception, and should speak with some other person in order to get back on track.
; 1/5 : Response employs relevant words and phrases, but does not demonstrate a sound understanding of the question or productive approaches to it. Student should seek assistance.
; 0/5 : No response, or response not relevant to the question.

==Weekly assignments==

* [[Math 361, Spring 2022, Assignment 1|Assignment 1]], due Wednesday, February 2.
* [[Math 361, Spring 2022, Assignment 2|Assignment 2]], due Wednesday, February 9.
* [[Math 361, Spring 2022, Assignment 3|Assignment 3]], due Wednesday, February 16.
* [[Math 361, Spring 2022, Assignment 4|Assignment 4]], due Wednesday, February 23.
* [[Math 361, Spring 2022, Assignment 5|Assignment 5]], due Wednesday, March 2.
* [[Math 361, Spring 2022, Assignment 6|Assignment 6]], due Wednesday, March 9.
* [[Math 361, Spring 2022, Assignment 7|Assignment 7]], due Wednesday, March 23.
* [[Math 361, Spring 2022, Assignment 8|Assignment 8]], due Wednesday, March 30.
* [[Math 361, Spring 2022, Assignment 9|Assignment 9]], due Wednesday, April 6.
* [[Math 361, Spring 2022, Assignment 10|Assignment 10]], due Wednesday, April 13.
* [[Math 361, Spring 2022, Assignment 11|Assignment 11]], due Wednesday, April 20.
* [[Math 361, Spring 2022, Assignment 12|Assignment 12]], due Wednesday, April 27.
* [[Math 361, Spring 2022, Assignment 13|Assignment 13]], due Wednesday, May 4.
* [[Math 361, Spring 2022, Assignment 14|Assignment 14]], due Wednesday, May 11.
* [[Math 361, Spring 2022, Assignment 15|Assignment 15]], due before final exam.

Math 360, Fall 2021, Assignment 15

2021-12-14T23:00:46Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully state the following theorems (you do not need to prove them):== # Corollary (of Lagrange's Theorem) concerning groups of prime order. # Corollary (of La..."

__NOTOC__
==Carefully state the following theorems (you do not need to prove them):==

# Corollary (of Lagrange's Theorem) concerning groups of prime order.
# Corollary (of Lagrange's Theorem) concerning $g^{\left\lvert G\right\rvert}$.

==Solve the following problems:==

# Describe all subgroups of $S_3$, and draw the subgroup diagram. ''(Hint: apart from the trivial and improper subgroups, the only possible orders are $2$ and $3$. Both of these are prime numbers.)''
# Describe all subgroups of $D_5$, and draw the subgroup diagram. ''(Use the same technique as in the previous problem.)''
# Show that, by contrast, $D_4$ has non-trivial proper subgroups which are ''not'' cyclic; hence, the technique of the previous two problems would have missed some subgroups.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 12

2021-12-14T18:15:46Z

Steven.Jackson: Created page with "__NOTOC__ ''"When I think of Euclid even now, I have to wipe my sweaty brow."'' : - C. M. Bellman ==Carefully define the following terms, then give one example and one non-e..."

__NOTOC__
''"When I think of Euclid even now, I have to wipe my sweaty brow."''

: - C. M. Bellman

==Carefully define the following terms, then give one example and one non-example of each:==

# Transposition (also known as a ''swap'').

==Carefully state the following theorems (you do not need to prove them):==

# Formula expressing any cycle as a product of transpositions.
# Theorem concerning generation of $S_n$ by transpositions ("Any subgroup of $S_n$ which contains all of the transpositions is...").

==Solve the following problems:==

# Express the cycle $(1,2,3,4,5)$ as a product of transpositions.
# Express the cycle $(2,3,4,5,1)$ as a product of transpositions.
# Observe that the cycles introduced in the previous two problems are in fact the same permutation. Is the decomposition of a permutation as a product of transpositions unique?
# (Optional) Skim through [https://en.wikipedia.org/wiki/Simplex the wikipedia article] on the geometric concept of a ''simplex,'' paying particular attention to the [https://en.wikipedia.org/wiki/Simplex#The_standard_simplex section on the "standard" simplex], which is ''regular'' in the sense described in the article. (For fun, see also the article on [https://en.wikipedia.org/wiki/5-cell the regular four-dimensional simplex], paying particular attention to the [https://en.wikipedia.org/wiki/5-cell#/media/File:5-cell.gif beautiful image] of a rotating projection of this object into three-dimensional space.) Finally, show that the symmetry group of the standard $n$-simplex is isomorphic to $S_{n+1}$. ''(Hint: for any pair (i,j), the map that swaps the $i$th and $j$th coordinates is an isometry of $\mathbb{R}^n$.)''
# Referring to the result of the above exercise, calculate the symmetry groups of the equilateral triangle, and of the regular tetrahedron. Do these results agree with our earlier findings?

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 14

2021-12-10T15:15:03Z

Steven.Jackson: Created page with "__NOTOC__ ''I must study politics and war, that my sons may have liberty to study mathematics and philosophy.'' : - John Adams, letter to Abigail Adams, May 12, 1780 ==Read:..."

__NOTOC__
''I must study politics and war, that my sons may have liberty to study mathematics and philosophy.''

: - John Adams, letter to Abigail Adams, May 12, 1780

==Read:==

# Section 10.

==Carefully define the following terms, then give one example and one non-example of each:==

# $\sim_{l,H}$ (the relation of ''left congruence modulo the subgroup $H$'').
# $\sim_{r,H}$ (the relation of ''right congruence modulo the subgroup $H$'').
# $xH$ (the ''left coset of $H$ by $x$).
# $Hx$ (the ''right coset of $H$ by $x$).
# $(G:H)$ (the ''index of $H$ in $G$;'' note that we did not discuss this in class, but it is Definition 10.13 in the text).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning a special property of kernels, not shared by arbitrary subgroups.
# Example to show that not every subgroup can be the kernel of a homomorphism.
# Theorem concerning the properties of the left congruence relation ("$\sim_{l,H}$ is an...").
# Theorem concerning the properties of the right congruence relation ("$\sim_{r,H}$ is an...").
# Theorem describing the elements of $xH$.
# Theorem describing the elements of $Hx$.
# Theorem relating left and right congruence when $G$ is abelian.
# Example to show that the result of the previous theorem may be false when $G$ is not abelian.
# Theorem relating the sizes of $xH$ and $yH$.
# Lagrange's Theorem.

==Solve the following problems:==

# Section 10, problems 1, 3, 6, 7, 9, 10, 12, 15, 20, 21, 22, 23, and 24.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 13

2021-12-03T22:06:32Z

Steven.Jackson: Created page with "__NOTOC__ ''"Reeling and Writhing, of course, to begin with,"'' ''the Mock Turtle replied;'' ''"And then the different branches of Arithmetic'' - ''Ambition, Distraction, Ugli..."

__NOTOC__
''"Reeling and Writhing, of course, to begin with,"'' ''the Mock Turtle replied;''
''"And then the different branches of Arithmetic'' - ''Ambition, Distraction, Uglification, and Derision."''

: - Lewis Carroll, ''Alice's Adventures in Wonderland''

==Read:==

# Section 13.

==Carefully define the following terms, then give one example and one non-example of each:==

# Even permutation.
# Odd permutation.
# $A_n$ (the ''alternating group on $n$ letters'').
# Homomorphism.
# Monomorphism.
# Epimorphism.
# Forward image (of a subset, under a function).
# Pre-image (of a subset, under a function).
# Pushforward (of a subgroup, under a homomorphism).
# Pullback (of a subgroup, under a homomorphism).
# Image (of a homomorphism).
# Kernel (of a homomorphism).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning the orbit count of $\tau\pi$, where $\tau$ is a transposition and $\pi$ is an arbitrary permutation.
# Theorem concerning the sign of $\tau\pi$, where $\tau$ is a transposition and $\pi$ is an arbitrary permutation.
# Formula for $\mathrm{sgn}(\tau_1\dots\tau_k)$, where $\tau_1,\dots,\tau_k$ are transpositions.
# Formula for $\mathrm{sgn}(\pi\sigma)$, where $\pi$ and $\sigma$ are arbitrary permutations.
# Formula for the order of $A_n$.
# Theorem characterizing the pushforward of a subgroup ("The pushforward of a subgroup is a...")
# Theorem characterizing the pullback of a subgroup ("The pullback of a subgroup is a...")

==Solve the following problems:==

# Section 13, problems 1, 2, 3, 8, 9, 10, 17, and 18.
# Determine whether the following permutations are odd or even: (i) $(12)$, (ii) $(123)$, and (iii) $(1234)$.
# Suppose $\sigma$ is an $l$-cycle, where $l$ is even. Is $\sigma$ an even permutation or an odd permutation?
# Suppose $\sigma$ is an $l$-cycle, where $l$ is odd. Is $\sigma$ an even permutation or an odd permutation?
# Define a function $f:\{1,2,3,4\}\rightarrow\{a,b,c,d\}$ as follows: $f(1)=a, f(2)=a, f(3)=b, f(4)=c$. Compute the following: (i) $f[f^{-1}[\{c,d\}]]$ and (ii) $f^{-1}[f[\{2,3\}]]$.
# Suppose $f:A\rightarrow B$ is a function, and $S\subseteq A$. Prove that $f^{-1}[f[S]]\supseteq S$. Give examples to show that equality may or may not hold.
# Suppose $f:A\rightarrow B$ is a function, and $T\subseteq B$. Prove that $f[f^{-1}[T]]\subseteq T$. Give examples to show that equality may or may not hold.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 10

2021-11-19T21:35:33Z

Steven.Jackson: Created page with "__NOTOC__ ''I have found a very great number of exceedingly beautiful theorems." : - Pierre de Fermat ==Carefully define the following terms, then give one example and one n..."

__NOTOC__
''I have found a very great number of exceedingly beautiful theorems."

: - Pierre de Fermat

==Carefully define the following terms, then give one example and one non-example of each:==

# Permutation model (for a group $G$).
# Finite permutation model (for a group $G$).

==Carefully state the following theorems (you do not need to prove them):==

# Cayley's Theorem.

==Solve the following problems:==

# Our proof of Cayley's Theorem was ''constructive,'' meaning that we proved that every group has a permutation model by actually constructing and exhibiting one specific permutation model. The particular permutation model constructed in our proof is called the ''Cayley model,'' or in some contexts, the ''left regular representation.'' Explicitly write down the Cayley model of the dihedral group $D_4$. (That is, explicitly write the permutation assigned by the Cayley model to each of the eight elements of $D_4$.)
# Now show that permutation models are not unique, by writing down a second and different permutation model for $D_4$. ''(Hint: begin by numbering the vertices of a square.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 11

2021-11-19T20:48:27Z

Steven.Jackson: Created page with "__NOTOC__ ''The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.'' : - Saint Au..."

__NOTOC__
''The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.''

: - Saint Augustine

==Read:==

# Section 9.

==Carefully define the following terms, then give one example and one non-example of each:==

# Fixed point (of a permutation $\pi$).
# Moving point (of a permutation $\pi$).
# Disjoint (permutations $\pi$ and $\sigma$).
# Orbit (of a permutation $\pi$).
# Cycle.
# $(i_1,\dots,i_k)$ (the ''cycle determined by the sequence $i_1,\dots,i_k$'').
# Length (of a cycle).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating $\sigma\tau$ to $\tau\sigma$, when $\sigma$ and $\tau$ are disjoint.
# Theorem concerning disjoint cycle decomposition.

==Carefully practice the following calculations, giving a worked example of each:==

# Conversion of two-row notation to cycle notation.
# Conversion of cycle notation to two-row notation.
# Composition of two permutations, expressed in cycle notation.
# Inversion of a permutation, expressed in cycle notation.

==Solve the following problems:==

# Section 9, problems 1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. ''(Hint for 14-16: recall that the order of a finite group element $g$ is the least positive integer $n$ with $g^n=1$. For a permutation, this will be the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 9

2021-11-12T21:04:58Z

Steven.Jackson:

__NOTOC__
''The moving power of mathematical invention is not reasoning but the imagination.''

: - Augustus de Morgan

==Read:==

# Section 8.

==Carefully define the following terms, then give one example and one non-example of each:==

# Permutation (of a set $S$).
# $\mathrm{Sym}(S)$ (the ''symmetric group on $S$'').
# $S_n$ (the ''symmetric group on $n$ letters'').
# Order (of a group; see the text for this definition).
# $D_n$ (the ''$n$th dihedral group;'' see the text).

==Carefully state the following theorems (you do not need to prove them):==

# Containment criterion for subgroups of $\mathbb{Z}_n$.
# Theorem relating $\left\langle[a]\right\rangle$ to $\left\langle[\mathrm{gcd}(a,n)]\right\rangle$ (in $\mathbb{Z}_n$).
# Classification of subgroups of $\mathbb{Z}_n$ ("Every subgroup of $\mathbb{Z}_n$ is generated by a unique...").
# Theorem concerning the order of $S_n$.

==Solve the following problems:==

# Section 6, problems 23, 25, and 27.
# Section 8, problems 1, 3, 5, 7, 9, 11, 13, 17, 30, 31, and 33.
# In class, we listed all permutations of the set $\{1,2,3\}$. Now, make a list of all permutations of the set $\{a,b,c\}$. Do you see any relationship between the two lists?
# Suppose $S$ and $T$ are sets, and that $f:S\rightarrow T$ is a bijection. Define a function $\phi:\mathrm{Sym}(S)\rightarrow\mathrm{Sym}(T)$ by the formula $\phi(\pi)=f\circ\pi\circ f^{-1}$. Show that $\phi(\pi\circ\sigma)=\phi(\pi)\circ\phi(\sigma)$.
# With notation as above, take $S=\{1,2,3\}$ and $T=\{a,b,c\}$, and let $f:S\rightarrow T$ be given by $f(1)=a, f(2)=b, f(3)=c$. For any specific $\pi\in\mathrm{Sym}(S)$ (e.g. for $\pi=\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}$), compute the permutation $\phi(\pi)\in\mathrm{Sym}(T)$. Do this in several more specific cases.
# Returning to the general case, define a new function $\psi:\mathrm{Sym}(T)\rightarrow\mathrm{Sym}(S)$ by the formula $\psi(\tau)=f^{-1}\circ\tau\circ f$. Show that $\phi\circ\psi$ and $\psi\circ\phi$ are both the identity maps, so that $\psi$ is the inverse of $\phi$. (Note in particular that this shows that $\phi$ is an invertible function and hence is a bijection.)
# Prove that $\phi$ is an isomorphism from $(\mathrm{Sym}(S),\circ)$ to $(\mathrm{Sym}(T),\circ)$.
# Finally, show that whenever $S$ and $T$ are equinumerous sets, $\mathrm{Sym}(S)$ and $\mathrm{Sym}(T)$ are isomorphic groups.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==
==Definitions:==
# Permutation (of a set $S$): Let $S$ be a set. A permutation of $S$ is a bijection from $S$ to itself.
# $\mathrm{Sym}(S)$ (the ''symmetric group on $S$''): $U(Fun(S, S), \circ) = (Sim(S), \circ)$. That is, $Sym(S)$ means the permutation of $S$, regarded as a group under function composition. It's called the symmetric group on $S$.
# $S_n$ (the ''symmetric group on $n$ letters''): $Sym(\{1,2,3 \cdots ,n\}) = S_n$.
# Order (of a group; see the text for this definition): the number of elements present in that group $|S|$.
# $D_n$ (the ''$n$th dihedral group;'' see the text): the group of symmetries of the regular n-gon.

==Theorem:==
# Containment criterion for subgroups of $\mathbb{Z}_n$: In $\mathbb Z_n, \left\langle [a] \right\rangle \subseteq \left\langle [b] \right\rangle \iff gcd(b,n) | a.$
# Theorem relating $\left\langle[a]\right\rangle$ to $\left\langle[\mathrm{gcd}(a,n)]\right\rangle$ (in $\mathbb{Z}_n$): In $\mathbb Z_n, \left\langle [a] \right\rangle = \left\langle [gcd(a,n)] \right\rangle$.
# Classification of subgroups of $\mathbb{Z}_n$ ("Every subgroup of $\mathbb{Z}_n$ is generated by a unique..."): Each subgroup has a unique generator (between $1$ and $n$) which divides $n$.
# Theorem concerning the order of $S_n$: $|S_n| = n!$.

==Book Problems:==
23. $<0>, <1>, <2>, <3>, <4>, <6>, <9>, <12>$ and $<18>$. (The subgroup
diagram is laid out similarly to Figure 6.18, with one subgroup above another
precisely when its generator divides the other's generator.)

25. $|<1>|=6/1=6, |<2>|=6/2=3, |<3>|=6/3=2, |<0>|=|<6>|=6/6=1$.

27. $|<1>|=12/1=12, |<2>|=12/2=6, |<3>|=12/3=4, |<4>|=12/4=3, |<6>|=12/6=2,
|<0>|=|<12>|=12/12=1.$

1. $\begin{pmatrix}1&2&3&4&5&6\\1&2&3&6&5&4\end{pmatrix}$

3. $\begin{pmatrix}1&2&3&4&5&6\\3&4&1&6&2&5\end{pmatrix}$

5. $\begin{pmatrix}1&2&3&4&5&6\\2&6&1&5&4&3\end{pmatrix}$

7. 2

9. $\mu^2=\iota$, so $\mu^{100}=(\mu^2)^{50}=\iota^{50}=\iota.$

In problems 11 and 13, the orbit of a point is the set of all points to which
it can be moved by repeated application of the permutation.

11. {1,2,3,4,5,6}.

13. {1,5}.

17. There are four choices for where to send $1$, only one choice for where to
send $2$, three choices for where to send $3$, two choices for where to send
$4$, and one choice for where to send $5$, so $4*1*3*2 = 24.$

30. Yes, this is a permutation. For injectivity note that
$f(x_1)=f(x_2)\implies x_1+1=x_2+1\implies x_1=x_2$, and for surjectivity note
that for any $y\in\mathbb{R}, f(y-1)=y$.

31. No, this map is not injective: $f(-1)=f(1)$ but $1\neq-1$.

33. Although this map is injective, it is not a permutation because it is not
surjective: its image consists of the ''positive'' reals, not all reals.

==Other Problems:==
# $\begin{pmatrix}a&b&c\\a&b&c\end{pmatrix}, \begin{pmatrix}a&b&c\\a&c&b\end{pmatrix}, \begin{pmatrix}a&b&c\\b&a&c\end{pmatrix}, \begin{pmatrix}a&b&c\\b&c&a\end{pmatrix}, \begin{pmatrix}a&b&c\\c&b&a\end{pmatrix}, \begin{pmatrix}a&b&c\\c&a&b\end{pmatrix}$. It seems as though writing down all of the permutations of $\{1,2,3\}$ and then systematically making the replacements $1\mapsto a, 2\mapsto b, 3\mapsto c$ results in a correct list of permutations of $\{a,b,c\}$. Perhaps this correspondence might even be a group isomorphism.
# $\phi(\pi\circ\sigma)=f\circ\pi\circ\sigma\circ f^{-1}$ while $\phi(\pi)\circ\phi(\sigma)=f\circ\pi\circ f^{-1}\circ f\circ\sigma\circ
f^{-1}$.
# $\begin{pmatrix}1&2&3\\a&b&c\end{pmatrix} \circ \begin{pmatrix}1&2&3\\3&2&1\end{pmatrix} = \begin{pmatrix}1&2&3\\c&b&a\end{pmatrix}$, $\begin{pmatrix}1&2&3\\c&b&a\end{pmatrix} \circ \begin{pmatrix}1&2&3\\a&b&c\end{pmatrix}^{-1}$ $= \begin{pmatrix}1&2&3\\c&b&a\end{pmatrix} \circ \begin{pmatrix}a&b&c\\1&2&3\end{pmatrix} = \begin{pmatrix}a&b&c\\c&b&a\end{pmatrix} \in Sym(T)$. By some miraculous coincidence, simply making the replacements $i\mapsto f(i)$ in the two-row notation for $\pi$ yields correct two-row notation for $\phi(\pi)$.
# For any permutation $\pi\in\mathrm{Sym}(T)$, we have $\phi(\psi(\pi))=\phi(f^{-1}\circ\pi\circ f)=f\circ f^{-1}\circ\pi\circ f\circ f^{-1}=\pi$. Similarly, for any $\sigma\in\mathrm{Sym}(S)$ we have $\psi(\phi(\sigma))=\sigma$. This shows that $\psi=\phi^{-1}$ and, in particular, that $\phi$ is a bijection.
# We showed above that $\phi$ is bijective and preserves operations, so it is an isomorphism.
# If $S$ and $T$ are equinumerous sets, then we can choose a bijection $f:S\rightarrow T$ and use it as above to produce an isomorphism $\phi:\mathrm{Sym}(S)\rightarrow\mathrm{Sym}(T)$.

Math 360, Fall 2021, Assignment 8

2021-11-12T20:04:30Z

Steven.Jackson:

__NOTOC__
''Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.''

: - John Locke, ''Second Reply to the Bishop of Worcester''

==Carefully define the following terms, then give one example and one non-example of each:==

# $\mathrm{gcd}(a,b)$.
# $\mathrm{lcm}(a,b)$.
# $|$ (the ''divisibility relation'' on $\mathbb{Z}$).

==Carefully state the following theorems (you need not prove them):==

# Classification of subgroups of cyclic groups ("Every subgroup of a cyclic group is...").
# Containment criterion for subgroups of $\mathbb{Z}$.
# Equality criterion for subgroups of $\mathbb{Z}$.
# Classification of subgroups of $\mathbb{Z}$ ("Every subgroup of $\mathbb{Z}$ has a unique...").
# Theorem concerning the properties of $\mathrm{gcd}(a,b)$.
# Theorem concerning the properties of $\mathrm{lcm}(a,b)$.

==Solve the following problems:==

# Section 6, problems 5, and 7.
# Compute $\mathrm{gcd}(6,9)$. Then compute $\mathrm{gcd}(-6,9)$. ''(Hint: think about the subgroups $\left\langle 6,9\right\rangle$ and $\left\langle -6,9\right\rangle$.)''
# For any $a,b\in\mathbb{Z}$, prove that $\left\langle a,b\right\rangle=\left\langle -a,b\right\rangle$. ''(Hint: prove mutual containment. Bear in mind that $\left\langle a,b\right\rangle$ is the ''smallest'' subgroup containing $a$ and $b$, so it is contained in any other subgroup that contains $a$ and $b$.)''
# Prove that for any $a,b\in\mathbb{Z}$, we have $\mathrm{gcd}(-a,b)=\mathrm{gcd}(a,b)$.
# Prove that for any $a,b\in\mathbb{Z}$, the set $S_{a,b}=\{xa+yb\,|\,x,y\in\mathbb{Z}\}$ is a subgroup of $\mathbb{Z}$.
# Prove that the subgroup $S_{a,b}$ above is in fact equal to $\left\langle a,b\right\rangle$. ''(Hint: prove mutual containment.)''
# Prove that for any integers $a,b\in\mathbb{Z}$, there exist integers $x,y$ with $xa+yb=\mathrm{gcd}(a,b)$.
# (Optional) Read [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm the Wikipedia article on the Extended Euclidean Algorithm], which is a very efficient computer algorithm that computes $\mathrm{gcd}(a,b)$ as well as the integers $x,y$ referenced above. This algorithm, which is lightning-fast even when the inputs $a,b$ are astronomically large, is foundational to many cryptographic and cryptanalytic techniques.
# (Optional) Implement the Extended Euclidean Algorithm in the programming language of your choice.
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==
==Definitions:==
# Suppose $a, b \in \mathbb Z$. the non-negative generator of $\left\langle a, b \right\rangle$ is denoted $gcd(a, b)$.
# $\left\langle a \right\rangle \cap \left\langle b \right\rangle$ is a subgroup of $\mathbb Z$, so it has a unique non-negative generator. This is denoted $lcm(a,b)$.
# Divisibility in $\mathbb Z$: $a|b$ ("a divides b" or "b is a multiple of a") means that $\exists c \in \mathbb Z$ such that $b = ac$.

==Theorems:==
#
# Containment criterion for subgroups of $\mathbb Z$): In $(\mathbb Z, +), \left\langle a \right\rangle \subseteq \left\langle b \right\rangle \iff b|a$.
# Equality criterion: $(\left\langle a \right\rangle = \left\langle b \right\rangle \iff b|a \wedge a|b)$. In $\mathbb Z, (a|b \text{ and } b|a) \iff b=\pm a$.
# Any subgroup of $\mathbb Z$ has a unique non-negative generator.
# (1) $gcd(a,b)$ is in fact a common divisor of $a \text{ and } b$: $gcd(a,b) |a \text{ and } gcd(a,b) | b$. (2) If $d$ is any other common divisor of $a, b$, then $d|gcd(a,b)$.
# (1) $lcm(a,b)$ is a common multiple of $a, b: a|lcm(a,b) \text{ and } b|lcm(a,b)$. (2)If $m$ is any other common multiple, then $lcm(a,b)|m$.

==Book Problems:==
5. $8$

7. $60$

==Problems:==
2. $gcd(6,9) = 3$, $gcd(6,-9) = 3$

3. $-a|a, b|b$, then $a \in \left\langle -a \right\rangle , b \in \left\langle b \right\rangle \rightarrow a \in \left\langle -a, b \right\rangle \wedge b \in \left\langle -a, b \right\rangle$. $\left\langle a, b \right\rangle$ is the smallest subgroup that contains $a, b$, $\left\langle -a, b \right\rangle$ also contains $a, b$, which means that $\left\langle a, b \right\rangle \subseteq \left\langle -a, b \right\rangle$. With similar proof we get $\left\langle -a, b \right\rangle \subseteq \left\langle a, b \right\rangle$. Therefore, $\left\langle a, b \right\rangle = \left\langle -a, b \right\rangle$

4. $\left\langle a, b \right\rangle = \left\langle -a, b \right\rangle$, $\left\langle a, b \right\rangle$ and $\left\langle -a, b \right\rangle$ have same generator. Generator of $\left\langle a, b \right\rangle = gcd(a, b)$, generator of $\left\langle -a, b \right\rangle = gcd(-a, b)$. Therefore, $gcd(a,b)=gcd(-a,b)$.

5. (a) $0a+0b = 0, \text{ identity } \in S_{alb}$; (2) $\forall x_1, y_1, x_2, y_2 \in \mathbb Z, if x_{1}a + y_{1}b \in S_{alb} \wedge x_{2}a + y_{2}b \in S_{alb}$, $(x_{1}a + y_{1}b + x_{2}a + y_{2}b) = (x_{1} + x_{2})a + (y_{1} + y_{2})b,(x_{1} + x-{2}) \in \mathbb Z, (y_{1} + y_{2}) \in \mathbb Z \rightarrow (x_{1}a + y_{1}b + x_{2}a + y_{2}b) \in S_{a, b} $. (3) Inverse $= -(xa + yb) = (-x)a + (-y)b, (-x), (-y) \in \mathbb Z$, inverse $\in S_{a, b}$. $S_{a, b}$ is a subgroup of $\mathbb Z$.

6. By taking $x=1, b=0$ we see that $a\in S_{a,b}$. Similarly, we also have $b\in S_{a,b}$. Since $\left\langle a,b\right\rangle$ is the ''smallest'' subgroup containing both $a$ and $b$, this gives $\left\langle a,b\right\rangle\subseteq S_{a,b}$. On the other hand, any element of $S_{a,b}$ must also be an element of $\left\langle a,b\right\rangle$ simply because $\left\langle a,b\right\rangle$ contains both $a$ and $b$ and is closed under addition and negation.

7. Since $\left\langle a,b\right\rangle=\left\langle\mathrm{gcd}(a,b)\right\rangle$ we certainly have $\mathrm{gcd}(a,b)\in\left\langle a,b\right\rangle$ and thus $\mathrm{gcd}(a,b)\in S_{a,b}$.

==Euclidean Code (Euclidean.java):==

import stdlib.StdOut;
public class Euclidean {
// Entry point.
public static void main(String[] args) {
int a = Integer.parseInt(args[0]);
int b = Integer.parseInt(args[1]);
StdOut.println(gcd(a, b));
}
// Returns the gcd
private static String gcd(int k1, int k2) {
// Set integer a to be the bigger number
int a = k1;
// Set integer b to be the smaller number
int b = k2;
if (k1 < k2) {
a = k2;
b = k1;
}
else if (k1 > k2) {
a = k1;
b = k2;
}
else if (k1 == k2) {
a = k1;
b = k2;
}
// quotient q
int q = 0;
// reminder r1 (final reminder)
int r1 = a;
// reminder r2 (temp reminder)
int r2 = b;
// coefficient for a, s1
int s1 = 1;
int s2 = 0;
// coefficient for b, t1
int t1 = 0;
int t2 = 1;
while (r2 > 0) {
q = r1/r2;
int temp_r = r2;
r2 = r1 - q * temp_r;
r1 = temp_r;
int temp_s = s2;
s2 = s1 - q * temp_s;
s1 = temp_s;
int temp_t = t2;
t2 = t1 - q * temp_t;
t1 = temp_t;
}
String s = r1 + " = " + "(" + s1 + ")" + "*" + "(" + a + ")" + " + " + "(" + t1 + ")" + "*" + "(" + b + ")";
return s;
}
}

Result:

jingwens-MBP:workspace jingwenfeng$ java Euclidean 240 46

2 = (-9)*(240) + (47)*(46)

Math 360, Fall 2021, Assignment 7

2021-11-12T19:37:27Z

Steven.Jackson:

__NOTOC__
''Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.''

: - Lord Kelvin

==Read:==

# Section 6.

==Carefully define the following terms, then give one example and one non-example of each:==

# Multiplicative notation (for a general group).
# Additive notation (for a general abelian group).
# Cyclic group.
# Generator (of a cyclic group).

==Carefully state the following theorems (you need not prove them):==

# Laws of exponents.
# Laws of multiples (i.e. the restatement of the laws of exponents in additive notation).
# Theorem concerning integer division.
# Classification of cyclic groups.

==Solve the following problems:==

# Section 6, problems 1, 3, 9, 10, 17, 19, 33, 34, 35, 36, and 37.
# Prove that every cyclic group is abelian. ''(Hint: every element has the form $g^i$ for some fixed generator $g$; now use the laws of exponents.)''
# Prove that every cyclic group is countable (i.e. either finite or countably infinite; you may utilize the classification of cyclic groups even though we have not yet completed its proof in class).
# Show that each of the following subgroups of $(\mathbb{Z},+)$ can be generated by a single non-negative integer: (a) $\left\langle 4, 6\right\rangle$, (b) $\left\langle 15, 35\right\rangle$, and (c) $\left\langle 12, 18, 27\right\rangle$.
# (Challenge) Following the pattern of the three parts of the last problem, try to guess a general formula for a single non-negative generator for the subgroup $\left\langle k_1,k_2,\dots,k_m\right\rangle$ of $(\mathbb{Z},+)$.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==
==Definitions:==
# ''Multiplicative notation'' is a notational system in which the operation symbol is suppressed and the operation is indicated by juxtaposition. Thus, for example, in the group $(G,\triangle)$, the quantity $a\triangle b$ is abbreviated to $ab$. In this system the identity element is denoted by the symbol $1$. Also, $a^n$ means $aa\dots a$ ($n$ factors) when $n>0$, it means $a'a'\dots a'$ ($\left\lvert n\right\rvert$ factors) when $n<0$, and it means $1$ when $n=0$.
# ''Additive notation'' is a notational system, reserved by convention for abelian groups, in which the operation is indicated by the symbol $+$. In this system the identity element is denoted $0$, and the quantity which would be written as $a^n$ in multiplicative notation is instead denoted $na$. # Suppose $G$ is a group. $G$ is said to be cyclic if there is some $g \in G$ with $\left\langle \{g\} \right\rangle = G$.
# Let $G$ be a cyclic group. A ''generator'' for $G$ is an element $g\in G$ with $\left\langle g\right\rangle=G$.

==Theorems:==
# Laws of Exponents: Suppose $G$ is any group, written multiplicatively. (1) $a^{i}a^{j} = a^{i+j}$, (2) $(a^{i})^{j} = a^{ij}$
# $(ia) + (ja) = (i+j)a, j(ia) = (ji)a$
# Suppose $a,b \in \mathbb Z$ and $b>0$. Then there exist unique $q, r \in \mathbb Z$, satisfying: $(1)a=bq+r, (2) 0 \leq r < b$.
# Every cyclic group is isomorphic to $(\mathbb Z, +)$ or to $(\mathbb Z_n, +_n)$ for some n.

==Textbook Solution:==

1. 42 = 9·4+6, q = 4, r = 6

3. −50 = 8(−7)+6, q = −7, r = 6

Using the structure theory of finite cyclic groups, developed in subsequent
assignments, it is not hard to prove that $[a]$ generates $\mathbb{Z}_n$ if
and only if $\mathrm{gcd}(a,n)=1$, and this fact provides a convenient check
for problems 9 and 10 below. However, it is not actually needed to answer
these questions: it is straightforward to simply calculate all of the cyclic
subgroups of the given groups and see which ones are improper. Similarly,
the structure theory of finite cyclic groups provides convenient shortcuts for
problem 17 but is not actually needed for it.

9. 1, 3, 5, and 7 are relatively prime to 8 so the answer is 4.

10. 1, 5, 7, and 11 are relatively prime to 12 so the answer is 4.

17. gcd(25, 30) = 5 and 30/5 = 6 so <25> has 6 elements, according to $g^{n}$.

19. The polar angle for i is π/2, so it generates a subgroup of 4 elements. $(e^{I\frac{\pi}{2}} = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}) = i)$

33. The Klein 4-group, D4

34. $(\mathbb R, +)$

35. $\mathbb{Z}_2$

36. No such example exists. Every infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$ which has just two generators, 1 and -1.

37. $\mathbb{Z}_8$ has generators 1, 3, 5, and 7.

==Solution:==

2. Let G be a cyclic group with a generator g∈G. Namely, we have G = ⟨g⟩ (every element in G is some power of g.). Let a and b be arbitrary elements in G. Then there exists n,m∈Z such that $a = g^n$ and $b = g^m$. It follows that

$ab = g^n g^m = g^{n+m} = g^{m+n} = g^m g^n = ba$

Hence we obtain ab = ba for arbitrary a,b∈G. Thus G is an abelian group.

3. Any cyclic group is isomorphic (hence also equinumerous) to either $\mathbb{Z}$ or to $\mathbb{Z}_n$, and both of these are countable.

4. For (a), note that $\left\langle 4,6\right\rangle=\{4x+6y\,|\,x,y\in\mathbb{Z}\}$, and any element of this set is evidently divisible by $2$, showing that $\left\langle4,6\right\rangle\subseteq\left\langle2\right\rangle$. On the other hand, $2=6-4$ is a member of $\left\langle4,6\right\rangle$, so we also have $\left\langle2\right\rangle\subseteq\left\langle4,6\right\rangle$ and thus in fact $\left\langle4,6\right\rangle=\left\langle2\right\rangle$. Similar arguments show that $\left\langle15,35\right\rangle=\left\langle5\right\rangle$ and $\left\langle12,18,27\right\rangle=\left\langle3\right\rangle$.

5. It would seem that $\left\langle k_1,\dots,k_n\right\rangle$ might be
generated by the greatest common divisor of $k_1,\dots,k_n$.

Math 360, Fall 2021, Assignment 6

2021-11-12T19:11:13Z

Steven.Jackson:

__NOTOC__
''I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.''

: - Thomas Mann, ''The Magic Mountain''

==Read:==

# Section 5.

==Carefully define the following terms, then give one example and one non-example of each:==

# Symmetry (of a subset $A\subseteq\mathbb{R}^n$).
# Symmetry group (of a subset $A\subseteq\mathbb{R}^n$).
# Order (of a group; see Definition 5.3 on page 50 of the text).
# $D_n$ (the ''dihedral group'' with order $2n$).
# Subgroup (of a group).
# Trivial subgroup (see Definition 5.5 on page 51 of the text).
# Improper subgroup (see Definition 5.5 on page 51 of the text).
# $\left\langle S\right\rangle$ (the ''subgroup generated by the subset $S$'').
# $\left\langle g\right\rangle$ (the ''cyclic subgroup generated by the element $g$''; see Definition 5.18 on page 54).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning unions and intersections of subgroups.

==Solve the following problems:==

# Section 5, problems 1, 2, 8, 9, 11, 12, 21, 22, 23, 24, 25, and 36.
# Prove that $(\mathbb{R},+)$ is ''not'' a cyclic group. ''(Hint: $\mathbb{R}$ is an uncountable set. Now look again at the list of elements of a cyclic subgroup. What can you conclude about the cardinality of a cyclic group?)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==
==Definitions:==
# Symmetry: the property that a mathematical object remains unchanged under a set of operations or transformations. Suppose $A \subseteq R^{n}$. A symmetry of $A$ is an isometry $\sigma : R^{n} \rightarrow R^{n}$ such that if $x \in A$ then also $\sigma (x) = A$. Claims: (1) The composition of two symmetries of $A$ is again a symmetry of $A$. (2) The identity map is always a symmetry of $A$ ("trivial symmetry"). (3) Any symmetry is invertible, and its inverse is again a symmetry.
# Symmetry group: Suppose $A\subseteq\mathbb{R}^n$. The ''symmetry group'' of $A$ is the set of symmetries of $A$ (see above), regarded as a group under composition. (Not to be confused with the ''symmetric group'' $\mathrm{Sym}(A)$, which is the group of all bijections from $A$ to $A$ and is usually a far larger group than the ''symmetry group'' defined here.)
# If $G$ is a group, then the order $|G|$ of $G$ is the number of elements in $G$. (Recall from Section 0 that, for any set S, $|S|$ is the cardinality of $S$.)
# The symmetry group of a regular n-gon.
# Subgroup: Suppose $(G, \triangle )$ is a group, and $H \subseteq G$. We say that $H$ is a subgroup of $G$ if (1) $e \in H$, (2) if $h_{1}, h_{2} \in H$, then $h_{1} \triangle h_{2} \in H$, (3) if $h \in H$, then $h' \in H$. (This ensure that $(H, \triangle )$ is also a group).
# If $G$ is a group, The subgroup $\{e\}$ is the trivial subgroup of G. All other subgroups are nontrivial.
# If $G$ is a group, then the subgroup consisting of $G$ itself is the improper subgroup of $G$. All other subgroups are proper subgroups.
# Suppose $(G, \triangle )$ is a group, and $S \subseteq G$ (subset, not necessarily a subgroup). define $\left\langle S \right\rangle$ to be the intersection of all subgroups of $G$ that contain $S$.
# let $G$ be a group and let $g \in G$. then the subgroup $\{ g^n | n\in \mathbb Z \}$ of $G$, characterized in Theorem 5.17, is called the cyclic subgroup of $G$ generated by $g$, and denote by $\left\langle g \right\rangle$. Theorem 5.17: $H = \{ g^n | n \in \mathbb Z \}$ is a subgroup of $G$ and is the smallest subgroup go $G$ that contain $g$, that is, every subgroup containing $g$ contains $H$.

==Theorems:==
# Suppose $(G, \triangle )$ is a group, and $H \leq G$ (is subgroup of) and $K \leq G$. Then $H \cap K$ is also a subgroup (however, $H \cup K$ is usually not).

==Book Problems:==
# 1. For $\mathbb R \in \mathbb C$, under addition, (1) $e = 0 \in \mathbb C$, $0 \in \mathbb R, e \in \mathbb R$. (2) If $a, b \in \mathbb R,a + b \in \mathbb R$. (3) If $a \in \mathbb R$, $a' = -a \mathbb R$. Yes, it is a group.
# 2. No. (1) $0 \not\in \mathbb Q^{+}$, (3)For $a \in \mathbb Q^{+}, a' = -a < 0 \not\in \mathbb Q^{+}$.
# 8. No. Suppose this is a set $A$. (1)$\det(e) = 1 \neq 2, e \not\in A$. Inverse not in it. (2)$det(M) * det(N) = 4 \neq 2, \forall M, N \in A$, not closed under matrix multiplication.
# 9. Yes. $I=\mathrm{diag}(1,1,\dots,1)$ is a diagonal matrix without zeros on the diagonal. If $A=\mathrm{diag}(a_1,\dots,a_n)$ and $B=\mathrm{diag}(b_1,\dots,b_n)$ are two such matrices, then so it $AB=\mathrm{diag}(a_1b_1,\dots,a_nb_n)$, as is $A^{-1}=\mathrm{diag}(a_1^{-1},\dots,a_n^{-1})$. (Note that although $\{(1, 1), (1, 2) \}$ has determinant $0$, is it not a diagonal matrix.)
# 11. No: the identity matrix does not belong to this set.
# 12. Yes: the identity matrix belongs to this set, and if $\det(A)=\pm1$ and $\det(B)=\pm1$ then also $\det(AB)=\pm1$ and $\det(A^{-1})=\pm1$.
# 21. (a) $\{ -50, -25, 0, 25, 50 \}$. (b) $\{4, 2, 1, \frac{1}{2}, \frac{1}{4} \}$. (c) $\{ \frac{1}{\pi ^2}, \frac{1}{\pi}, 1, \pi, \pi^2 \}$.
# 22. $\left\{\begin{bmatrix}0&-1\\-1&0\end{bmatrix},\begin{bmatrix}1&0\\0&1\end{bmatrix}\right\}$.
# 23. $\left\{\left.\begin{bmatrix}1&n\\0&1\end{bmatrix}\,\right\rvert\,n\in\mathbb{Z}\right\}$.
# 24. $\left\{\left.\begin{bmatrix}3^n&0\\0&2^n\end{bmatrix}\,\right\rvert\,n\in\mathbb{Z}\right\}$.
# 25. $\left\{\dots,\begin{bmatrix}0&-1/2\\-1/2&0\end{bmatrix},\begin{bmatrix}1&0\\0&1\end{bmatrix},\begin{bmatrix}0&-2\\-2&0\end{bmatrix},\begin{bmatrix}4&0\\0&4\end{bmatrix},\begin{bmatrix}0&-8\\-8&0\end{bmatrix},\dots\right\}$.
# 36. (a) see http://jwilson.coe.uga.edu/EMAT6680/Parsons/MVP6690/Essay1/modular.html (b) $\{0\}, \mathbb Z_6, \{0, 2, 4\}, \{0, 3\}, \{0, 2, 4\}, \mathbb Z_6$. (c)$1, 5$ (d)See https://drive.google.com/file/d/1ZU7k9peVMdC_mY5XCtIT5-Ld8Z4k_bjy/view?usp=sharing

==Other problems:==
By the classification of cyclic groups, and cyclic group must be isomorphic with either $(\mathbb{Z},+)$ or with $(\mathbb{Z}_n,+_n)$. In particular, any cyclic group must be equinumerous either with $\mathbb{Z}$ or with $\mathbb{Z}_n$. But $\mathbb{R}$ is an uncountable set, so $(\mathbb{R},+)$ cannot be cyclic.

Math 360, Fall 2021, Assignment 9

2021-11-05T02:21:00Z

Steven.Jackson: Created page with "__NOTOC__ ''The moving power of mathematical invention is not reasoning but the imagination.'' : - Augustus de Morgan ==Read:== # Section 8. ==Carefully define the followi..."

Math 360, Fall 2021, Assignment 8

2021-10-29T18:26:46Z

Steven.Jackson: Created page with "__NOTOC__ ''Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.'' : - John Locke, ''Second Reply to the Bish..."

Math 360, Fall 2021, Assignment 6

2021-10-22T20:46:35Z

Steven.Jackson: Created page with "__NOTOC__ ''I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.'' : - Thomas M..."

Math 360, Fall 2021, Assignment 7

2021-10-22T16:59:51Z

Steven.Jackson: Created page with "__NOTOC__ ''Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.'' : - Lord Kelvin ==Read:=..."

Math 360, Fall 2021, Assignment 5

2021-10-08T01:15:49Z

Steven.Jackson: Created page with "__NOTOC__ ''I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fair..."

__NOTOC__
''I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.''

: - Jonathan Swift, ''Gulliver's Travels''

==Read:==

# Section 3.

==Carefully define the following terms, then give one example and one non-example of each:==

# Substructure (of a binary structure).
# Unit (in a monoid).
# $\mathcal{U}(M)$ (the ''group of units'' of a monoid $(M,\triangle)$).
# Isomorphism (from one binary structure to another).
# Isomorphic (binary structures).
# Structural property.

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning the place of $\mathcal{U}(M)$ in the "hierarchy of niceness" (i.e. whether it is necessarily a semigroup, a monoid, a group, and/or an abelian group).

==Solve the following problems:==

# Section 3, problems 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 29, 30, 31, and 32.
# '''(Rigid motions)''' Recall that $\mathbb{R}^n$ denotes the set of all ordered $n$-tuples of real numbers. Given two points $\vec{x}=(x_1,\dots,x_n)$ and $\vec{y}=(y_1,\dots,y_n)$, the distance between these points is given by the ''distance formula'' $d(\vec{x},\vec{y})=\sqrt{(x_1-y_1)^2+\dots+(x_n-y_n)^2}$. An ''isometry of $\mathbb{R}^n$'' (also known as a ''rigid motion of $\mathbb{R}^n$'') is a bijection $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with the property that for any pair of points $\vec{x},\vec{y}\in\mathbb{R}^n$, one has $d(f(\vec{x}),f(\vec{y}))=d(\vec{x},\vec{y})$. For example, the function $f:\mathbb{R}^1\rightarrow\mathbb{R}^1$ given by $f((x_1))=(x_1+3)$ is an isometry, since it is bijective and $\sqrt{((x_1+3)-(y_1+3))^2}=\sqrt{((x_1-y_1)^2)}$. On the other hand, the motion $g((x_1))=(x_1^3)$ is not rigid even though it is bijective since, for instance, the points $(1)$ and $(2)$ lie at distance $1$ but their images under $g$ lie at distance $7$. Give as many examples as you can of rigid motions of $\mathbb{R}^2$, and then give examples of motions of $\mathbb{R}^2$ that are ''not'' rigid.
# '''($\mathrm{Iso}(\mathbb{R}^n)$)''' Let $\mathrm{Iso}(\mathbb{R}^n)$ denote the set of all isometries of $\mathbb{R}^n$. Prove that $\mathrm{Iso}(R^n)$ is a substructure of $(\mathrm{Fun}(\mathbb{R}^n,\mathbb{R}^n),\circ)$. ''(Hint: you only need to show that the composition of two isometries is an isometry. This is easier than it looks.)''
# (Challenge) Prove that $(\mathrm{Iso}(\mathbb{R}^n),\circ)$ is in fact a group. ''(Hint: it is relatively straightforward to show that this structure is associative and has an identity. Once that is done, it remains to show that the inverse of an isometry is again an isometry. Some cleverness is required for the last part, though once the proof is begun in the right way, it is quite short.)''
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 4

2021-10-01T20:52:34Z

Steven.Jackson: Created page with "__NOTOC__ ''Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.'' : - Goethe..."

__NOTOC__
''Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.''

: - Goethe

==Read:==

# Section 4.

==Carefully define the following terms, then give one example and one non-example of each:==

# $(\mathrm{Fun}(S,S))$.
# $f\circ g$ (the ''composition'' of the functions $f$ and $g$).
# $\iota$ (the ''identity function'' from a set $S$ to itself).
# $\mathbb{Z}_n$.
# $+_n$ (addition modulo $n$).
# $\cdot_n$ (multiplication modulo $n$).
# Semigroup.
# Monoid.
# Inverse (of an element of a monoid).
# Group.
# Abelian (group).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem regarding associativity of composition.
# Theorem asserting that $+_n$ is well-defined.

==Solve the following problems:==

# Section 4, problems 1, 3, 5, 11, 12, 13, 14, 18, and 19.
# Give an example of (a) a binary structure which is not a semigroup, (b) a semigroup which is not a monoid, (c) a monoid which is not a group, and (d) a group which is not abelian.
# By making an operation table, determine which elements of the commutative monoid $(\mathbb{Z}_5,\cdot_5)$ have inverses. Then do the same for $(\mathbb{Z}_6,\cdot_6)$ and $(\mathbb{Z}_8,\cdot_8)$.
# Based on the results of the previous problem, try to make a conjecture regarding which elements of $(\mathbb{Z}_n,\cdot_n)$ have inverses.
# Carefully show that the operation $\cdot_n$ is well-defined (i.e. state and prove a theorem analogous to the theorem asserting that $+_n$ is well-defined).

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021

2021-10-01T13:07:22Z

Steven.Jackson:

==Course information==

* See the [http://cartan.math.umb.edu/classes/f21_ma360/f21_ma360_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Tuesdays and Thursdays, 4:00 p.m.-5:15 p.m., in W-1-34.
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: W-3-154-27
* Office hours: Tuesdays and Thursdays, 3:00 p.m.-3:50 p.m., and Wednesdays, 1:00 p.m.-1:50 p.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].
* Telephone: (617) 287-6469.

==Important dates==

* Weekly quizzes happen on Tuesdays during the last ten minutes of class. The first quiz is on Tuesday, September 14.
* First midterm: Tuesday, October 19.
* Second midterm: Tuesday, November 16.
* Final exam: Thursday, December 16, 3:00 p.m. - 6:00 p.m., in W-1-34.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Scoring rubric==

Quizzes and exam questions are all scored on a five-point scale, defined as
follows:

; 5/5 : Response demonstrates substantial mastery of the ideas assessed by the question. May contain small imperfections addressed in comments. Student should move forward and learn new things.
; 4/5 : Response demonstrates understanding of sound technique, but execution errors lead to wrong answer.
; 3/5 : Response is generally on the right track; student would probably solve the problem given sufficient time, but is not yet demonstrating full understanding of the ideas assessed by the problem. Student should spend more time in order to achieve full understanding.
; 2/5 : Response indicates a substantial misconception. Student is unlikely to make progress without first correcting the misconception, and should speak with some other person in order to get back on track.
; 1/5 : Response employs relevant words and phrases, but does not demonstrate a sound understanding of the question or productive approaches to it. Student should seek assistance.
; 0/5 : No response, or response not relevant to the question.

==Weekly assignments==

* [[Math 360, Fall 2021, Assignment 1|Assignment 1]], due Tuesday, September 14.
* [[Math 360, Fall 2021, Assignment 2|Assignment 2]], due Tuesday, September 21.
* [[Math 360, Fall 2021, Assignment 3|Assignment 3]], due Tuesday, September 28.
* [[Math 360, Fall 2021, Assignment 4|Assignment 4]], due Tuesday, October 5.
* [[Math 360, Fall 2021, Assignment 5|Assignment 5]], due Tuesday, October 12.
* [[Math 360, Fall 2021, Assignment 6|Assignment 6]], due Tuesday, October 19.
* [[Math 360, Fall 2021, Assignment 7|Assignment 7]], due Tuesday, October 26.
* [[Math 360, Fall 2021, Assignment 8|Assignment 8]], due Tuesday, November 2.
* [[Math 360, Fall 2021, Assignment 9|Assignment 9]], due Tuesday, November 9.
* [[Math 360, Fall 2021, Assignment 10|Assignment 10]], due Tuesday, November 16.
* [[Math 360, Fall 2021, Assignment 11|Assignment 11]], due Tuesday, November 23.
* [[Math 360, Fall 2021, Assignment 12|Assignment 12]], due Tuesday, November 30.
* [[Math 360, Fall 2021, Assignment 13|Assignment 13]], due Tuesday, December 7.
* [[Math 360, Fall 2021, Assignment 14|Assignment 14]], due Tuesday, December 14.
* [[Math 360, Fall 2021, Assignment 15|Assignment 15]], due before final exam.

Math 360, Fall 2021, Assignment 3

2021-09-24T02:28:06Z

Steven.Jackson: Created page with "__NOTOC__ ''We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater,..."

__NOTOC__
''We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.''

: - Voltaire

==Read:==

# Section 2.

==Carefully define the following terms, then give one example and one non-example of each:==

# Injective (function; a.k.a. ''one-to-one'' function).
# Surjective (function; a.k.a. ''onto'' function).
# Bijective (function).
# Equinumerous (sets).
# Countable (set).
# Uncountable (set).
# Binary operation (on a set $S$).
# Binary structure.
# Commutative (binary structure).
# Associative (binary structure).
# Left identity element (in a binary structure).
# Right identity element (in a binary structure).
# Identity element (in a binary structure).
# Invertible element (in a binary structure with identity).
# Inverse (of an element of a binary structure with identity).

==Carefully state the following theorems (you do not need to prove them):==

# Cantor's Theorem.
# Theorem bounding the number of two-sided identity elements in one binary structure.

==Solve the following problems:==

# Section 2, problems 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, and 23.
# Give an example of a binary structure $(S,\triangle)$ which has two distict left identity elements. ''(Hint: let $S$ be a set with a very small number of elements, e.g. $S=\{a,b,c\}$. Define your very own binary operation $\triangle$ on $S$ by means of a table, as discussed on page 24 of the text. Remember that you are the author of the table and may complete it however you wish, as long as the result is a legitimate binary structure. Within a short time, you are likely to see several ways of completing such a table which lead to multiple left identity elements.)''
# Using the method of the previous exercise, give an example of a binary structure with two distinct right identity elements.
# Now try to use the method of the last two exercises to produce a binary structure with two distinct (two-sided) identity elements. What goes wrong?
# Show that the closed interval $[0,1]$ of the real line is equinumerous with the closed interval $[0,2]$, by constructing an explicit bijection between these two sets. Then formally verify that your map is a bijection.
# Consult a calculus book for a graph of the function $f(x)=\tan^{-1}(x)$. Assuming that the graph is not misleading, explain why the whole of the real number system $\mathbb{R}$ is equinumerous with the open interval $\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$. ''(Hint: you may need to look up the ''horizontal line test'' for injectivity of functions from $\mathbb{R}$ to $\mathbb{R}$, and you may also need to think about how to determine the image of such a function by inspecting its graph.)''

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021

2021-09-17T20:56:29Z

Steven.Jackson:

==Course information==

* See the [http://cartan.math.umb.edu/classes/f21_ma360/f21_ma360_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Tuesdays and Thursdays, 4:00 p.m.-5:15 p.m., in W-1-34.
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: W-3-154-27
* Office hours: Tuesdays and Thursdays, 3:00 p.m.-3:50 p.m., and Wednesdays, 1:00 p.m.-1:50 p.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].
* Telephone: (617) 287-6469.

==Important dates==

* Weekly quizzes happen on Tuesdays during the last ten minutes of class. The first quiz is on Tuesday, September 14.
* First midterm: Tuesday, October 19.
* Second midterm: Tuesday, November 16.
* Final exam: To be announced.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Scoring rubric==

Quizzes and exam questions are all scored on a five-point scale, defined as
follows:

; 5/5 : Response demonstrates substantial mastery of the ideas assessed by the question. May contain small imperfections addressed in comments. Student should move forward and learn new things.
; 4/5 : Response demonstrates understanding of sound technique, but execution errors lead to wrong answer.
; 3/5 : Response is generally on the right track; student would probably solve the problem given sufficient time, but is not yet demonstrating full understanding of the ideas assessed by the problem. Student should spend more time in order to achieve full understanding.
; 2/5 : Response indicates a substantial misconception. Student is unlikely to make progress without first correcting the misconception, and should speak with some other person in order to get back on track.
; 1/5 : Response employs relevant words and phrases, but does not demonstrate a sound understanding of the question or productive approaches to it. Student should seek assistance.
; 0/5 : No response, or response not relevant to the question.

==Weekly assignments==

* [[Math 360, Fall 2021, Assignment 1|Assignment 1]], due Tuesday, September 14.
* [[Math 360, Fall 2021, Assignment 2|Assignment 2]], due Tuesday, September 21.
* [[Math 360, Fall 2021, Assignment 3|Assignment 3]], due Tuesday, September 28.
* [[Math 360, Fall 2021, Assignment 4|Assignment 4]], due Tuesday, October 5.
* [[Math 360, Fall 2021, Assignment 5|Assignment 5]], due Tuesday, October 12.
* [[Math 360, Fall 2021, Assignment 6|Assignment 6]], due Tuesday, October 19.
* [[Math 360, Fall 2021, Assignment 7|Assignment 7]], due Tuesday, October 26.
* [[Math 360, Fall 2021, Assignment 8|Assignment 8]], due Tuesday, November 2.
* [[Math 360, Fall 2021, Assignment 9|Assignment 9]], due Tuesday, November 9.
* [[Math 360, Fall 2021, Assignment 10|Assignment 10]], due Tuesday, November 16.
* [[Math 360, Fall 2021, Assignment 11|Assignment 11]], due Tuesday, November 23.
* [[Math 360, Fall 2021, Assignment 12|Assignment 12]], due Tuesday, November 30.
* [[Math 360, Fall 2021, Assignment 13|Assignment 13]], due Tuesday, December 7.
* [[Math 360, Fall 2021, Assignment 14|Assignment 14]], due Tuesday, December 14.
* [[Math 360, Fall 2021, Assignment 15|Assignment 15]], due before final exam.

Math 360, Fall 2021, Assignment 2

2021-09-17T20:35:07Z

Steven.Jackson: Created page with "__NOTOC__ ''No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. O..."

__NOTOC__
''No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....''

: - Morris Kline, ''Mathematics in Western Culture''

==Carefully define the following terms, then give one example and one non-example of each:==

# Binary relation (from $A$ to $B$).
# Reflexive (binary relation).
# Symmetric (binary relation).
# Anti-symmetric (binary relation).
# Transitive (binary relation).
# Equivalence relation.
# Equivalence class (of an element $a\in A$, with respect to an equivalence relation $\sim$ on $A$; also known as $\left[a\right]_\sim$).
# Partition (of a set $A$).
# $S/\sim$ (the ''partition arising from the equivalence relation $\sim$ on the set $S$'').
# $\equiv_n$ (the relation of congruence modulo a non-negative integer $n$).
# Function (from $A$ to $B$).
# Domain (of a function).
# Codomain (of a function).
# Image (of a function).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem relating equivalence relations to partitions.
# Theorem concerning the key properties of $\equiv_n$ (i.e. "$\equiv_n$ is an...").

==Solve the following problems:==

# Section 0, problems 12, 23, 25, 29, 30, 31, 32, and 33.
# Calculate the cardinality (i.e. the number of elements) of $\mathbb{Z}/\equiv_n$. Illustrate your calculation with a concrete example, listing the elements of $\mathbb{Z}/\equiv_n$ explicitly.
# A binary relation which is reflexive, anti-symmetric, and transitive is called a ''partial ordering.'' Give at least one example of a partial ordering. ''(Hint: you may wish to ignore the word "partial," which functions here mainly as a distraction.)''
# Now looks for an example of a partial ordering which shows why we should call them ''partial'' orderings in general.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 1

2021-09-10T13:54:00Z

Steven.Jackson:

__NOTOC__
''By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.''

: - Mary Shelley, ''Frankenstein''

==Read:==

# Section 0.

==Carefully define the following terms, then give one example and one non-example of each:==

# Containment (of sets).
# Equality (of sets).
# Property.
# Ordered pair.
# Cartesian product (of two sets).

==Carefully state the following theorems (you do not need to prove them):==

# Russell's paradox (this is not really a theorem, but it is an important fact).
# Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$).

==Solve the following problems:==

# Section 0, problems 1, 2, 3, 4, 5, 6, 7, 8, and 11.
# Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. ''(Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)''
# Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. ''(Hint: use the previous result together with the definition of set equality.)''
# The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on Russell's Paradox and its resolutions, see [https://en.wikipedia.org/wiki/Universal_set Wikipedia: Universal Set].)

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 360, Fall 2021, Assignment 1

2021-09-10T13:40:03Z

Steven.Jackson: Created page with "__NOTOC__ ''By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and al..."

__NOTOC__
''By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.''

: - Mary Shelley, ''Frankenstein''

==Read:==

# Section 0.

==Carefully define the following terms, then give one example and one non-example of each:==

# Containment (of sets).
# Equality (of sets).
# Property.
# Ordered pair.
# Cartesian product (of two sets).

==Carefully state the following theorems (you do not need to prove them):==

# Russel's paradox (this is not really a theorem, but it is an important fact).
# Basic counting principle (relating the size of $A\times B$ to the sizes of $A$ and $B$).

==Solve the following problems:==

# Section 0, problems 1, 2, 3, 4, 5, 6, 7, 8, and 11.
# Prove that for any set $S$, it must be the case that $\emptyset\subseteq S$. ''(Hint: begin with "Suppose not. Then by the definition of set containment, there must be some member of $\emptyset$ which is not a member of $S$. But...".)''
# Now suppose $S$ is any set with $S\subseteq\emptyset$. Prove that $S=\emptyset$. ''(Hint: use the previous result together with the definition of set equality.)''
# The previous two exercises show that $\emptyset$ is the "smallest of all sets." Is there a "largest of all sets?" (For more information on this question as well as on Russel's Paradox and its resolutions, see [https://en.wikipedia.org/wiki/Universal_set Wikipedia: Universal Set].)

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Main Page

2021-09-06T21:33:55Z

Steven.Jackson:

__NOTOC__
==Course Pages==

===Fall 2021===

* [[Math 360, Fall 2021|Math 360]], Abstract Algebra I

===Spring 2021===

* [[Math 361, Spring 2021|Math 361]], Abstract Algebra II

===Fall 2020===

* [[Math 360, Fall 2020|Math 360]], Abstract Algebra I

===Spring 2020===

* [[Math 361, Spring 2020|Math 361]], Abstract Algebra II

===Fall 2019===

* [[Math 260, Fall 2019|Math 260]], Linear Algebra I
* [[Math 360, Fall 2019|Math 360]], Abstract Algebra I

===Spring 2019===

* [[Math 242, Spring 2019|Math 242]], Multivariable and Vector Calculus
* [[Math 361, Spring 2019|Math 361]], Abstract Algebra II

===Fall 2018===

* [[Math 260, Fall 2018|Math 260]], Linear Algebra I
* [[Math 360, Fall 2018|Math 360]], Abstract Algebra I

===Spring 2018===

* [[Math 361, Spring 2018|Math 361]], Abstract Algebra II
* [[Math 380, Spring 2018|Math 380]], Introduction to Computational Algebraic Geometry

===Fall 2017===

* [[Math 260, Fall 2017|Math 260]], Linear Algebra I
* [[Math 360, Fall 2017|Math 360]], Abstract Algebra I

===Spring 2017===

* [[Math 260, Spring 2017|Math 260]], Linear Algebra I
* [[Math 361, Spring 2017|Math 361]], Abstract Algebra II

===Fall 2016===

* [[Math 360, Fall 2016|Math 360]], Abstract Algebra I
* [[Math 480, Fall 2016|Math 480]], Information Theory

===Spring 2016===

* [[Math 141, Spring 2016|Math 141]], Calculus II
* [[Math 361, Spring 2016|Math 361]], Abstract Algebra II

===Fall 2015===

* [[Math 260, Fall 2015|Math 260]], Linear Algebra I
* [[Math 360, Fall 2015|Math 360]], Abstract Algebra I

===Spring 2015===

* [[Math 361, Spring 2015|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2015|Math 480]], Introduction to Cryptography

===Fall 2014===

* [[Math 360, Fall 2014|Math 360]], Abstract Algebra I
* [[Math 440, Fall 2014|Math 440]], General Topology

===Spring 2014===

* [[Math 361, Spring 2014|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2014|Math 480]], Introduction to Computational Algebraic Geometry II

===Fall 2013===

* [[Math 242, Fall 2013|Math 242]], Calculus III
* [[Math 360, Fall 2013|Math 360]], Abstract Algebra I

===Spring 2013===

* [[Math 361, Spring 2013|Math 361]], Abstract Algebra II
* [[Math 480, Spring 2013|Math 480]], Introduction to Computational Algebraic Geometry

==Helpful Links==

* Summary of the [http://en.wikipedia.org/wiki/Help:Wiki_markup Mediawiki markup language].
* Tips on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

Math 360, Fall 2021

2021-09-06T21:31:40Z

Steven.Jackson: Created page with "==Course information== * See the [http://cartan.math.umb.edu/classes/f21_ma360/f21_ma360_syllabus.pdf syllabus] for general information and the schedule of readings. * Class..."

==Course information==

* See the [http://cartan.math.umb.edu/classes/f21_ma360/f21_ma360_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Tuesdays and Thursdays, 4:00 p.m.-5:15 p.m., in W-1-34.
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: W-3-154-27
* Office hours: Tuesdays and Thursdays, 3:00 p.m.-3:50 p.m., and Wednesdays, 1:00 p.m.-1:50 p.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].
* Telephone: (617) 287-6469.

==Important dates==

* Weekly quizzes happen on Tuesdays during the last ten minutes of class. The first quiz is on Tuesday, September 14.
* First midterm: Tuesday, October 19.
* Second midterm: Tuesday, November 16.
* Final exam: To be announced.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Weekly assignments==

* [[Math 360, Fall 2021, Assignment 1|Assignment 1]], due Tuesday, September 14.
* [[Math 360, Fall 2021, Assignment 2|Assignment 2]], due Tuesday, September 21.
* [[Math 360, Fall 2021, Assignment 3|Assignment 3]], due Tuesday, September 28.
* [[Math 360, Fall 2021, Assignment 4|Assignment 4]], due Tuesday, October 5.
* [[Math 360, Fall 2021, Assignment 5|Assignment 5]], due Tuesday, October 12.
* [[Math 360, Fall 2021, Assignment 6|Assignment 6]], due Tuesday, October 19.
* [[Math 360, Fall 2021, Assignment 7|Assignment 7]], due Tuesday, October 26.
* [[Math 360, Fall 2021, Assignment 8|Assignment 8]], due Tuesday, November 2.
* [[Math 360, Fall 2021, Assignment 9|Assignment 9]], due Tuesday, November 9.
* [[Math 360, Fall 2021, Assignment 10|Assignment 10]], due Tuesday, November 16.
* [[Math 360, Fall 2021, Assignment 11|Assignment 11]], due Tuesday, November 23.
* [[Math 360, Fall 2021, Assignment 12|Assignment 12]], due Tuesday, November 30.
* [[Math 360, Fall 2021, Assignment 13|Assignment 13]], due Tuesday, December 7.
* [[Math 360, Fall 2021, Assignment 14|Assignment 14]], due Tuesday, December 14.
* [[Math 360, Fall 2021, Assignment 15|Assignment 15]], due before final exam.

Math 361, Spring 2021, Assignment 15

2021-05-13T22:22:18Z

Steven.Jackson: Created page with "__NOTOC__ ==Carefully state the following theorems (you do not need to prove them):== # Classification of simple field extensions. ==Solve the following problems:== # Find..."

__NOTOC__

==Carefully state the following theorems (you do not need to prove them):==

# Classification of simple field extensions.

==Solve the following problems:==

# Find all real solutions of the equation $x^2-5x+3=0$. Be sure to find the ''exact'' solutions. Call these solutions $r_1$ and $r_2$.
# Compute the sum $r_1+r_2$, the product $r_1r_2$, and the squares $r_1^2$ and $r_2^2$. Do not make any approximations.
# Find a monomorphism $\phi$ from the quotient ring $\mathbb{Q}[x]/\left\langle x^2-13\right\rangle$ into the real number system.
# Show that the solutions $r_1,r_2$ that you found above both lie in the image of the monomorphism $\phi$ that you defined above, and compute the pre-images $\rho_1=\phi^{-1}(r_1)$ and $\rho_2=\phi^{-1}(r_2)$.
# Working in the quotient ring $\mathbb{Q}[x]/\left\langle x^2-13\right\rangle$, compute the sum $\rho_1+\rho_2$, the product $\rho_1\rho_2$, and the squares $\rho_1^2$ and $\rho_2^2$.
# If you needed to program a computer to store and manipulate the real solutions of the equation $x^2-5x+3=0$, what approach would you take?

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2021

2021-05-13T21:59:13Z

Steven.Jackson:

==Course information==

* See the [http://cartan.math.umb.edu/classes/s21_ma361/s21_ma361_syllabus.pdf syllabus] for general information and the schedule of readings.
* Class meets Mondays, Wednesdays, and Fridays, 11:00 a.m.-11:50 a.m., via [https://umassboston.zoom.us/j/92011627592 Zoom].
* Textbook: John Fraleigh, ''A First Course in Abstract Algebra,'' Seventh Edition.
* Instructor: [http://www.math.umb.edu/~jackson Steven Jackson].
* Office: [https://umassboston.zoom.us/j/92011627592 Zoom].
* Office hours: Mondays, Wednesdays, and Fridays, 10:00 a.m.-10:50 a.m.
* E-mail: [mailto:Steven.Jackson@umb.edu Steven.Jackson@umb.edu].

==Important dates==

* Weekly quizzes happen on Mondays during the last ten minutes of class. The first quiz is on Monday, February 1.
* First midterm: Monday, March 1.
* Second midterm: Monday, April 12.
* Final exam: Monday, May 17, 11:30 a.m.-2:30 p.m.

==Recorded classes==

Video recordings of our past classes are available to registered students via [https://umb.umassonline.net/webapps/blackboard/content/listContentEditable.jsp?content_id=_1326069_1&course_id=_22123_1&content_id=_1326409_1 Blackboard].

==Quiz link==

Weekly quizzes happen on Mondays, starting at 11:40 a.m., and are due by 11:50 a.m. (During weeks when Monday is a holiday, the quiz happens on Wednesday.) They are hosted on a [https://umb.umassonline.net/webapps/blackboard/content/listContentEditable.jsp?content_id=_1329797_1&course_id=_22123_1&mode=reset folder within our Blackboard site]. Although the quiz will not become visible until 11:40, you are free to sign in a few minutes before this, and I recommend doing this so that you will have the full ten minutes available to answer the quiz question. I also recommend that you remain signed in to our Zoom session while taking the quiz. I will remain in the meeting in case you have any questions.

==Exam link==

There is a [https://umb.umassonline.net/webapps/blackboard/content/listContentEditable.jsp?content_id=_1329900_1&course_id=_22123_1&mode=reset separate Blackboard folder for exams]. On exam days, please sign in to Zoom at least five minutes early with your video on, make sure you have photo ID available to hold up to the camera if requested, and click over to the Blackboard folder in another tab. The exam will appear at the official start time of the class. You will work each question on paper, then scan or photograph your work and upload the resulting image file as your response. More detailed exam instructions will be sent by e-mail as the exam dates approach.

==How to use this page==

Below you will find links to the weekly assignment pages. Each of these pages is editable by anyone in the class, so apart from telling you what problems to work on they are excellent spaces in which to ask questions. (If you are very shy you may ask your questions privately, either by [mailto:Steven.Jackson@umb.edu email] or in person. But we will all work more efficiently if you ask them on the wiki, so that each question only needs to be answered once.) It is also extremely helpful to try to answer questions posed by other students. I will monitor these pages to ensure that no wrong answers go uncorrected.

If you are not already familiar with them, you may wish to read about [http://en.wikipedia.org/wiki/Help:Wiki_markup wiki markup] and [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics]. Also, you may wish to add this page and the assignment pages to your [[Special:Watchlist|watchlist]] using the link in the upper right corner of each page, then change your [[Special:Preferences|preferences]] to enable e-mail notifications; this way you will know about page activity without constantly re-checking all the pages.

==Weekly assignments==

* [[Math 361, Spring 2021, Assignment 1|Assignment 1]], due Monday, February 1.
* [[Math 361, Spring 2021, Assignment 2|Assignment 2]], due Monday, February 8.
* [[Math 361, Spring 2021, Assignment 3|Assignment 3]], due Wednesday, February 17.
* [[Math 361, Spring 2021, Assignment 4|Assignment 4]], due Monday, February 22.
* [[Math 361, Spring 2021, Assignment 5|Assignment 5]], due Monday, March 1.
* [[Math 361, Spring 2021, Assignment 6|Assignment 6]], due Monday, March 8.
* [[Math 361, Spring 2021, Assignment 7|Assignment 7]], due Monday, March 22.
* [[Math 361, Spring 2021, Assignment 8|Assignment 8]], due Monday, March 29.
* [[Math 361, Spring 2021, Assignment 9|Assignment 9]], due Monday, April 5.
* [[Math 361, Spring 2021, Assignment 10|Assignment 10]], due Monday, April 12.
* [[Math 361, Spring 2021, Assignment 11|Assignment 11]], due Wednesday, April 21.
* [[Math 361, Spring 2021, Assignment 12|Assignment 12]], due Monday, April 26.
* [[Math 361, Spring 2021, Assignment 13|Assignment 13]], due Monday, May 3.
* [[Math 361, Spring 2021, Assignment 14|Assignment 14]], due Monday, May 10.
* [[Math 361, Spring 2021, Assignment 15|Assignment 15]], due before final exam.

Math 361, Spring 2021, Assignment 14

2021-05-06T21:54:53Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 29. ==Carefully define the following terms, and give one example and one non-example of each:== # Field extension. # Base field (of a field e..."

__NOTOC__

==Read:==

# Section 29.

==Carefully define the following terms, and give one example and one non-example of each:==

# Field extension.
# Base field (of a field extension $F\rightarrow E$).
# Extension field (of a field extension $F\rightarrow E$).
# Injection (of a field extension $F\rightarrow E$).
# Algebraic element (of a field extension $F\rightarrow E$).
# Transcendental element (of a field extension $F\rightarrow E$).
# $\mathrm{ann}_F(e)$ (the ''annihilator'' of $e$ over $F$).
# $\mathrm{irr}(e,F)$ (the ''minimal polynomial'' of $e$ over $F$).
# $\mathrm{deg}(e,F)$ (the ''degree'' of $e$ over $F$; we did not discuss this in class, but it is part of Definition 29.14 in the text).

==Carefully state the following theorems (you do not need to prove them):==

# Theorem concerning the annihilator of an element ("$\mathrm{ann}_F(e)$ is always an...")

==Solve the following problems:==

# Section 29, problems 1, 3, 5, 7, 9, 10, and 14.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2021, Assignment 13

2021-04-30T14:07:39Z

Steven.Jackson: Created page with "__NOTOC__ ==Read:== # Section 26. # Section 27. ==Carefully define the following terms, and give one example and one non-example of each:== # Maximal ideal. # Principal id..."

__NOTOC__

==Read:==

# Section 26.
# Section 27.

==Carefully define the following terms, and give one example and one non-example of each:==

# Maximal ideal.
# Principal ideal domain.
# Prime ideal (we did not discuss this in class; it is Definition 27.13 in the text).

==Carefully state the following theorems (you do not need to prove them):==

# Lemma regarding ideals which contain units.
# Theorem relating maximal ideals to fields.
# Containment criterion for principal ideals.
# Equality criterion for principal ideals.
# Theorem concerning ideals of $\mathbb{Z}$ (i.e. "$\mathbb{Z}$ is a...").
# Theorem concerning ideals of $F[x]$ (i.e. "$F[x]$ is a...").
# Theorem relating maximal ideals to irreducible elements in principal ideal domains.
# Theorem relating prime ideals to integral domains (we did not discuss this in class; it is Theorem 27.15 in the text).
# Theorem relating maximal ideals to prime ideals (we did not discuss this in class; it is Corollary 27.16 in the text).

==Solve the following problems:==

# Section 26, problem 24. ''(Hint: Since $F$ is a field and is isomorphic to $F/\{0\}$, we know that $\{0\}$ is a maximal ideal; from this we can write a list of all the ideals of $F$.)''
# Section 27, problems 1, 2, 3, 5, 7, 9, and 18.
# Find an irreducible polynomial of degree $4$ in $\mathbb{Z}_2[x]$. ''(Hint: this can be done with the Sieve, but there is a shorter way. First make a list of all irreducible polynomials of degree $2$; these are just the quadratics with no roots, and there is a simple pattern that predicts which polynomials will have roots in $\mathbb{Z}_2$. Next, write down a degree $4$ polynomial which has no roots, and test it for divisibility by the irreducible quadratics. If your root-free quartic is divisible by an irreducible quadratic, throw it away and try another root-free quartic, continuing until you find a root-free quartic which is not divisible by any irreducible quadratic.)''
# Construct a field with exactly sixteen elements. ''(Hint: the result of the previous question is directly relevant to this one.)'' Prove that your ring really is a field, ''without'' writing down the whole multiplication table.

<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==