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Math 361, Spring 2022, Assignment 15

2022-05-15T21:36:39Z

Jingwen.feng001: /* Revision: */

__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:==

==Definitions and Theorems:==
https://drive.google.com/file/d/1vFWDTJtosWx0XMEh8ZzTTCTQqueqkB-3/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1alSd3CfsFCsDpyeG1-lxmxfFGLz60iIF/view?usp=sharing

==Revision Notes:==

https://drive.google.com/file/d/1kXciJ1X10AMPXPK3m_ymzX6L7NjYpvO9/view?usp=sharing

Math 361, Spring 2022, Assignment 15

2022-05-15T21:35:56Z

Jingwen.feng001: /* Problems: */

__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:==

==Definitions and Theorems:==
https://drive.google.com/file/d/1vFWDTJtosWx0XMEh8ZzTTCTQqueqkB-3/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1alSd3CfsFCsDpyeG1-lxmxfFGLz60iIF/view?usp=sharing

==Revision:==

https://drive.google.com/file/d/1kXciJ1X10AMPXPK3m_ymzX6L7NjYpvO9/view?usp=sharing

Math 361, Spring 2022, Assignment 15

2022-05-15T04:12:58Z

Jingwen.feng001: /* Definitions and Theorems: */

Math 361, Spring 2022, Assignment 15

2022-05-15T03:05:01Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 12

2022-05-14T21:03:13Z

Jingwen.feng001: /* Solutions: */

__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:==

https://drive.google.com/file/d/1NnrPxxpnghVN8SEQrPYKcx5fvqlxecrH/view?usp=sharing

Math 361, Spring 2022, Assignment 14

2022-05-13T02:18:38Z

Jingwen.feng001: /* Solutions: */

__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:==
==Notes:==
https://drive.google.com/file/d/1m34inTUsMS_QGoX0qSQAuv2HvFz9jSE1/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/188n4cWhYW3Lx0R0nvxGL9r1920J9XWJM/view?usp=sharing

Math 361, Spring 2022, Assignment 13

2022-05-01T17:46:00Z

Jingwen.feng001: /* Solutions: */

__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:==
https://drive.google.com/file/d/1ocLy6EETSU9QL0UPaxyWx-6dx5EC1nkm/view?usp=sharing

Math 361, Spring 2022, Assignment 12

2022-04-30T21:20:08Z

Jingwen.feng001: /* Question: */

Math 361, Spring 2022, Assignment 11

2022-04-24T20:15:52Z

Jingwen.feng001: /* Solutions: */

__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:==
#Associate class (of an element of a domain $D$).

# Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).

# Characterization of the associate class $[a]_\sim$

# List of units in $\mathbb{Z}$.

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/17jnBWbZgvWYUD5UW7kqw6aGYCj-BebE-/view?usp=sharing

==Problems:==

https://drive.google.com/file/d/1cQz6Su4YeTzI2dH4OGu8FfFq3fGUIE8H/view?usp=sharing

Math 361, Spring 2022, Assignment 12

2022-04-24T20:12:26Z

Jingwen.feng001: Created page with "==Question:== Working once more in a general integral domain $D$, prove that if $a$ is irreducible and $a\sim b$, then $b$ is also irreducible."

==Question:==
Working once more in a general integral domain $D$, prove that if $a$ is irreducible and $a\sim b$, then $b$ is also irreducible.

Math 361, Spring 2022, Assignment 11

2022-04-19T16:52:53Z

Jingwen.feng001: /* Solutions: */

__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:==
#Associate class (of an element of a domain $D$).

# Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).

# Characterization of the associate class $[a]_\sim$

# List of units in $\mathbb{Z}$.

==Solutions:==
https://drive.google.com/file/d/1pFE9SV4pOqmF2F3quwKwkMlXIMkAeRQm/view?usp=sharing

Math 361, Spring 2022, Assignment 11

2022-04-18T20:49:13Z

Jingwen.feng001: /* Questions: */

Math 361, Spring 2022, Assignment 11

2022-04-18T20:48:49Z

Jingwen.feng001: /* Questions: */

Math 361, Spring 2022, Assignment 11

2022-04-18T20:48:19Z

Jingwen.feng001: /* Questions: */

Math 361, Spring 2022, Assignment 10

2022-04-18T19:06:48Z

Jingwen.feng001: /* Solutions: */

__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:==

==Definitions and Theorems:==
https://drive.google.com/file/d/12qg0ua83mJJKQduzpgLjo7SsZ2KGwvEl/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1sWc5CPO9jzdwHJEcU71iZCacFi1Sm7jf/view?usp=sharing

Math 361, Spring 2022, Assignment 9

2022-04-12T23:53:35Z

Jingwen.feng001: /* Cheatsheet Exam1+2: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1khlUPgHwDXH4DhX5snKTPhQsE68Cqxws/view?usp=sharing

==Problems:==

https://drive.google.com/file/d/1z0eT75OCSKfGq-QD7ANUnUZ2J4yZgFc9/view?usp=sharing

==Cheatsheet Exam1+2:==

https://drive.google.com/file/d/1XWXw2OHDhfGSbfX7UbdJreEjFTS8p8bL/view?usp=sharing

Math 361, Spring 2022, Assignment 9

2022-04-12T03:59:26Z

Jingwen.feng001: /* Problems: */

Math 361, Spring 2022, Assignment 9

2022-04-04T03:29:26Z

Jingwen.feng001: /* Definitions and Theorems: */

Math 361, Spring 2022, Assignment 9

2022-04-03T23:15:32Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 8

2022-03-28T13:48:29Z

Jingwen.feng001: /* Solutions: */

__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:==

==Definitions and Theorems:==

https://drive.google.com/file/d/1I90oqjlNhy0-M74UodNAEpQSBakoUiCO/view?usp=sharing

==Problems:==

Math 361, Spring 2022, Assignment 5

2022-03-21T03:37:18Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1cLuOX4boxHCGcHELYCEBDPo4CzWRB4aI/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1Yk8_-QaAj1WLzKfPoF6sM8xyof5zf_bs/view?usp=sharing

Math 361, Spring 2022, Assignment 6

2022-03-20T21:23:49Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/15F-NUWtjYWj_V5s1ZixPGkJgpyIoKfrG/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1aO3LiY-6kUvdmm8c2150t7D8s0QaIArN/view?usp=sharing

Math 361, Spring 2022, Assignment 7

2022-03-20T21:23:02Z

Jingwen.feng001: /* Problems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1wpXrdvbl5EfYppZSkzrSvpAZYpd-y-sj/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1kMmwxaFrWX53GKx0Flw1Tvg8U7cM4FNZ/view?usp=sharing

Prime.java (for generate primes p,q and public and private keys)

https://drive.google.com/file/d/1-jB3muBNNGE82hC5a_51x4i0W8y9kbDY/view?usp=sharing

javac Prime.java

java Prime 5

p = 23

q = 17

Math 361, Spring 2022, Assignment 7

2022-03-20T21:22:49Z

Jingwen.feng001: /* Problems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1wpXrdvbl5EfYppZSkzrSvpAZYpd-y-sj/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1kMmwxaFrWX53GKx0Flw1Tvg8U7cM4FNZ/view?usp=sharing

Prime.java (for generate primes p,q and public and private keys)
https://drive.google.com/file/d/1-jB3muBNNGE82hC5a_51x4i0W8y9kbDY/view?usp=sharing

javac Prime.java

java Prime 5

p = 23

q = 17

Math 361, Spring 2022, Assignment 7

2022-03-20T20:38:49Z

Jingwen.feng001: /* Problems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1wpXrdvbl5EfYppZSkzrSvpAZYpd-y-sj/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1IGRBFs2tMemL4MOI_uSVrioXG3lalDhM/view?usp=sharing

Prime.java (for generate primes p,q and public and private keys)
https://drive.google.com/file/d/1-jB3muBNNGE82hC5a_51x4i0W8y9kbDY/view?usp=sharing

javac Prime.java

java Prime 5

p = 23

q = 17

Math 361, Spring 2022, Assignment 7

2022-03-20T20:38:28Z

Jingwen.feng001: /* Problems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1wpXrdvbl5EfYppZSkzrSvpAZYpd-y-sj/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1IGRBFs2tMemL4MOI_uSVrioXG3lalDhM/view?usp=sharing

Prime.java (for generate primes p,q and public and private keys)
https://drive.google.com/file/d/1-jB3muBNNGE82hC5a_51x4i0W8y9kbDY/view?usp=sharing

java Prime 5

p = 23

q = 17

Math 361, Spring 2022, Assignment 7

2022-03-20T20:38:20Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1wpXrdvbl5EfYppZSkzrSvpAZYpd-y-sj/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1IGRBFs2tMemL4MOI_uSVrioXG3lalDhM/view?usp=sharing

Prime.java (for generate primes p,q and public and private keys)
https://drive.google.com/file/d/1-jB3muBNNGE82hC5a_51x4i0W8y9kbDY/view?usp=sharing

java Prime 5
p = 23
q = 17

Math 361, Spring 2022, Assignment 7

2022-03-19T20:18:20Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 6

2022-03-16T22:58:05Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 5

2022-03-16T20:00:13Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 4

2022-03-02T14:25:33Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/16HpHcJuiyCkTaeRBan3IsfSl1WO6qCY6/view?usp=sharing

Exam Preparation:
https://drive.google.com/file/d/1YWB5VtKh7FsHVT8_SBlD0yQ2N4Ylmc1E/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/15RrqY0HgCub1eHbvhjdIkyzLSdOWZTIi/view?usp=sharing

Math 361, Spring 2022, Assignment 4

2022-03-02T03:48:31Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/16HpHcJuiyCkTaeRBan3IsfSl1WO6qCY6/view?usp=sharing

Exam Preparation:
https://drive.google.com/file/d/1VZ2ebH1X8dLmwJzls7SinidzOX92XCXS/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/15RrqY0HgCub1eHbvhjdIkyzLSdOWZTIi/view?usp=sharing

Math 361, Spring 2022, Assignment 4

2022-03-01T20:46:34Z

Jingwen.feng001: /* Problems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/16HpHcJuiyCkTaeRBan3IsfSl1WO6qCY6/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/15RrqY0HgCub1eHbvhjdIkyzLSdOWZTIi/view?usp=sharing

Math 361, Spring 2022, Assignment 4

2022-03-01T20:45:30Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/16HpHcJuiyCkTaeRBan3IsfSl1WO6qCY6/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1hpKxgsRKqobpOX2J0-3GjwBkh4ww9tRI/view?usp=sharing

Math 361, Spring 2022, Assignment 4

2022-02-27T03:36:44Z

Jingwen.feng001: /* Problems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1-z_gvaAAuES4R8xNYpxj2XIBO0CGGwhO/view?usp=sharing

==Problems:==
https://drive.google.com/file/d/1hpKxgsRKqobpOX2J0-3GjwBkh4ww9tRI/view?usp=sharing

Math 361, Spring 2022, Assignment 4

2022-02-27T03:36:37Z

Jingwen.feng001: /* Definitions and Theorems: */

__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:==
Definitions:
# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).
Theorems:
# 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...").

==Solutions:==
==Definitions and Theorems:==
https://drive.google.com/file/d/1-z_gvaAAuES4R8xNYpxj2XIBO0CGGwhO/view?usp=sharing

==Problems:==

Math 361, Spring 2022, Assignment 4

2022-02-26T19:58:32Z

Jingwen.feng001: /* Definitions and Theorems: */

Math 361, Spring 2022, Assignment 4

2022-02-26T19:57:33Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 4

2022-02-26T19:57:17Z

Jingwen.feng001: /* Questions: */

Math 361, Spring 2022, Assignment 4

2022-02-26T19:57:05Z

Jingwen.feng001: /* Questions: */

__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:==

Definitions:

# $R/I$ (the ''quotient'' of the ring $R$ by the two-sided ideal $I$).

Theorems:

# 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...").

==Solutions:==

Math 361, Spring 2022, Assignment 4

2022-02-26T19:56:34Z

Jingwen.feng001: /* Questions: */

__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:==
Theorems:

# 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...").

==Solutions:==

Math 361, Spring 2021, Assignment 4

2022-02-23T03:40:03Z

Jingwen.feng001: /* Carefully state the following theorems (you do not need to prove them): */

__NOTOC__

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

# $R/I$ (the ''quotient'' of the ring $R$ by the ideal $I$).
# $\pi:R\rightarrow R/I$ (the ''canonical projection'' of $R$ onto $R/I$).
# $\mathbb{C}$ (the complex number system, which we have defined as a certain quotient ring).
# $i$ (the so-called "imaginary" number whose square is $-1$).

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

# Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is..."): $\bigstar$ Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is..."): When $I$ is an ideal, Multiplication of cosets in $R/I$ is well-defined.
# Fundamental Theorem of Ring Homomorphisms: Let $\phi : G \rightarrow G'$ be a group homomorphism with kernel $H$. Then $\phi[G]$ is a group, and $\mu : G/H \rightarrow \phi[G]$ given by $\mu(gH) = \phi(g)$ is an isomorphism. If $\gamma : G \rightarrow G/H $is the homomorphism given by $\gamma(g) = gH,$ then $\phi(g) = \mu\gamma(g)$ for each $g \in G$.

==Solve the following problems:==

# (Complex inversion) Let $a+bi$ be any element of $\mathbb{C}$ ''other than'' $0_{\mathbb{C}}$ (i.e. assume that $a$ and $b$ are not both zero). Prove that $a+bi$ is a unit, with inverse $\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i$.
# Suppose that $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_3$ is a ring homomorphism satisfying $\phi(1)=(1,1)$. Compute $\phi(2)$. ''(Hint: $\phi(2)=\phi(1+1)$.)''
# Let $\phi$ be as in the previous problem. Compute a table of values for $\phi$, showing enough lines to get a clear sense of the pattern in the outputs of $\phi$.
# With $\phi$ as above, describe $\ker(\phi)$.
# With $\phi$ as above, describe $\mathrm{im}(\phi)$.
# Prove that $\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. ''(Hint: use the results of the previous problems, together with the Fundamental Theorem of Ring Homomorphisms.)''
# Repeat all of the above exercises except the last, this time supposing that $\phi$ is a ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}_2\times\mathbb{Z}_2$ satisfying $\phi(1)=(1,1)$.
# We will see next week that the ring $\mathbb{Z}_2\times\mathbb{Z}_2$ is ''not'' isomorphic to $\mathbb{Z}_4$. Why does this not contradict the results of the previous exercises?
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2021, Assignment 4

2022-02-23T03:39:05Z

Jingwen.feng001: /* Carefully state the following theorems (you do not need to prove them): */

__NOTOC__

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

# $R/I$ (the ''quotient'' of the ring $R$ by the ideal $I$).
# $\pi:R\rightarrow R/I$ (the ''canonical projection'' of $R$ onto $R/I$).
# $\mathbb{C}$ (the complex number system, which we have defined as a certain quotient ring).
# $i$ (the so-called "imaginary" number whose square is $-1$).

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

# Theorem concerning the definition of coset multiplication (i.e. "Multiplication of cosets in $R/I$ is well-defined if $I$ is...").
# Fundamental Theorem of Ring Homomorphisms: Let $\phi : G \rightarrow G'$ be a group homomorphism with kernel $H$. Then $\phi[G]$ is a group, and $\mu : G/H \rightarrow \phi[G]$ given by $\mu(gH) = \phi(g)$ is an isomorphism. If $\gamma : G \rightarrow G/H $is the homomorphism given by $\gamma(g) = gH,$ then $\phi(g) = \mu\gamma(g)$ for each $g \in G$.

==Solve the following problems:==

# (Complex inversion) Let $a+bi$ be any element of $\mathbb{C}$ ''other than'' $0_{\mathbb{C}}$ (i.e. assume that $a$ and $b$ are not both zero). Prove that $a+bi$ is a unit, with inverse $\frac{a}{a^2+b^2}-\frac{b}{a^2+b^2}i$.
# Suppose that $\phi:\mathbb{Z}\rightarrow\mathbb{Z}_2\times\mathbb{Z}_3$ is a ring homomorphism satisfying $\phi(1)=(1,1)$. Compute $\phi(2)$. ''(Hint: $\phi(2)=\phi(1+1)$.)''
# Let $\phi$ be as in the previous problem. Compute a table of values for $\phi$, showing enough lines to get a clear sense of the pattern in the outputs of $\phi$.
# With $\phi$ as above, describe $\ker(\phi)$.
# With $\phi$ as above, describe $\mathrm{im}(\phi)$.
# Prove that $\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. ''(Hint: use the results of the previous problems, together with the Fundamental Theorem of Ring Homomorphisms.)''
# Repeat all of the above exercises except the last, this time supposing that $\phi$ is a ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}_2\times\mathbb{Z}_2$ satisfying $\phi(1)=(1,1)$.
# We will see next week that the ring $\mathbb{Z}_2\times\mathbb{Z}_2$ is ''not'' isomorphic to $\mathbb{Z}_4$. Why does this not contradict the results of the previous exercises?
<center><big>'''--------------------End of assignment--------------------'''</big></center>

==Questions:==

==Solutions:==

Math 361, Spring 2022, Assignment 1

2022-02-14T05:33:49Z

Jingwen.feng001: /* Solutions: */

__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:==
Complete Notes:

https://drive.google.com/file/d/1wCXbcBMd8br--G0xReIPzf8FN9sTtfHU/view?usp=sharing

==Definitions:==
# $H\leq G$: $H$ is a subgroup of $G$.
# $H\trianglelefteq G$: H is a normal subgroup of G. (When $\sim_{l,H}$ and $\sim_{r,H}$ are the same relation: if $\forall g \in G, \forall h \in H, ghg^{-1} \in H$
# Coset multiplication (when $H\trianglelefteq G):(g_{1}H)(g_{2}H) = (g_{1}g_{2})H$.
# Canonical projection (from $G$ onto $G/H$): Suppose $H\trianglelefteq G$. Define $\pi : G \rightarrow G/H$ as follows: $\pi (a) = aH$. Then $\pi$ is an epimorphism.

==Theorems:==
# Theorem characterizing when coset multiplication is well-defined: If H is a normal subgroup of G, Then coset multiplication is well-defined.
# Theorem concerning the properties of coset multiplication ("When $H\trianglelefteq G$, coset multiplication turns $G/H$ into a..."): If $H\trianglelefteq G$, then $G/H$ is a group under coset multiplication.
# Theorem describing the kernel of the canonical projection: Let $\pi : G \rightarrow G/H$ be the canonical projection. Then $ker(\pi) = H$.
# Fundamental Theorem of Homomorphisms: Suppose $\phi : G \rightarrow H$ is a homomorphism. Let $\phi : G \rightarrow G/Ker(\phi)$ be the canonical projection. Then there exists a unique monomorphism $\widehat{\phi}: G/Ker(\phi) \rightarrow H$ such that $\widehat{\phi} \circ \pi = \phi$

==Problems:==
1. $0+ \left\langle 4 \right\rangle$ = $\{0, 4, 8\}; 1+ \left\langle 4 \right\rangle$ = $\{1, 5, 9\} \cdots$. operation table: A table of $\mathbb Z_4$.

Book Problem 1. $\mathbb Z_6 /\left\langle 3 \right\rangle$ $\left\langle 3 \right\rangle = \{0, 3\}$. Order $= \frac{6}{2} = 3$

Book Problem 9. 4

Book Problem 24. 2, $\mathbb Z_2$

Book Problem 30. $|G/H| = m, (aH)^{m} = eH = H, (aH)^{m} = (a^m)H, a^m \in H$.

Book Problem 31. $k \in K$, $K$ is the intersection of all normal subgroups. $gkg^{-1} \in$ all normal subgroups, therefore, $gkg^{-1} \in$ intersection of all normal subgroups, $gkg^{-1} \in K$. K is a normal subgroup.

3. Yes, 4

4. yes, 2

5. $\pi (0) = 0 + \left\langle 4 \right\rangle, \pi (1) = 1 + \left\langle 4 \right\rangle, \pi (2) = 2 + \left\langle 4 \right\rangle, \pi (3) = 3 + \left\langle 4 \right\rangle$

6. $aE = {a}, bE = {b}, aE = bE, {a}={b}, a == b$

7. $\forall g_1 \in G, g_2 \in G, g_1 g_2 g_1^{-1} \in G$ because of closure. $G/G: \{G\}$

Math 361, Spring 2022, Assignment 3

2022-02-13T05:13:07Z

Jingwen.feng001: /* Problems: */

__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:==
Complete Notes:

https://drive.google.com/file/d/1biJbwmyG2FefR-eiZ0QOFmcEOG8aDLud/view?usp=sharing

==Definitions:==
# Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.
# Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
# Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
# Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $ a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
# Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
# Zero (a.k.a. ''trivial'') subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
# Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
# Subring generated by a subset.
# Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

==Theorems:==
# Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
# Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
# Theorem relating fields to integral domains: Every field is an integral domain.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

==Problems:==
Answer to problems:

https://drive.google.com/file/d/1ieuGx7P7BYDkEANSxwteao4HwZjswYVx/view?usp=sharing

Java Program for the book problems 1, 2, 3, 4:

https://drive.google.com/file/d/1qhd2Kw4f1v164GG3qfh5Q9nVqbGA59xe/view?usp=sharing

Math 361, Spring 2022, Assignment 3

2022-02-13T05:12:19Z

Jingwen.feng001: /* Problems: */

__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:==
Complete Notes:

https://drive.google.com/file/d/1biJbwmyG2FefR-eiZ0QOFmcEOG8aDLud/view?usp=sharing

==Definitions:==
# Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.
# Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
# Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
# Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $ a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
# Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
# Zero (a.k.a. ''trivial'') subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
# Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
# Subring generated by a subset.
# Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

==Theorems:==
# Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
# Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
# Theorem relating fields to integral domains: Every field is an integral domain.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

==Problems:==
Answer to problems:
https://drive.google.com/file/d/1ieuGx7P7BYDkEANSxwteao4HwZjswYVx/view?usp=sharing

Java Program I used for the book problems:
https://drive.google.com/file/d/1qhd2Kw4f1v164GG3qfh5Q9nVqbGA59xe/view?usp=sharing

Math 361, Spring 2022, Assignment 3

2022-02-13T05:09:45Z

Jingwen.feng001: /* Problems: */

__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:==
Complete Notes:

https://drive.google.com/file/d/1biJbwmyG2FefR-eiZ0QOFmcEOG8aDLud/view?usp=sharing

==Definitions:==
# Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.
# Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
# Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
# Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $ a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
# Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
# Zero (a.k.a. ''trivial'') subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
# Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
# Subring generated by a subset.
# Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

==Theorems:==
# Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
# Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
# Theorem relating fields to integral domains: Every field is an integral domain.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

==Problems:==
Answer to problems:
https://drive.google.com/file/d/1ieuGx7P7BYDkEANSxwteao4HwZjswYVx/view?usp=sharing

Math 361, Spring 2022, Assignment 3

2022-02-13T05:08:23Z

Jingwen.feng001: /* Problems: */

__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:==
Complete Notes:

https://drive.google.com/file/d/1biJbwmyG2FefR-eiZ0QOFmcEOG8aDLud/view?usp=sharing

==Definitions:==
# Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.
# Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
# Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
# Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $ a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
# Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
# Zero (a.k.a. ''trivial'') subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
# Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
# Subring generated by a subset.
# Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

==Theorems:==
# Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
# Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
# Theorem relating fields to integral domains: Every field is an integral domain.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

==Problems:==
https://drive.google.com/file/d/1ieuGx7P7BYDkEANSxwteao4HwZjswYVx/view?usp=sharing

Math 361, Spring 2022, Assignment 3

2022-02-13T05:08:08Z

Jingwen.feng001: /* Solutions: */

Math 361, Spring 2022, Assignment 3

2022-02-13T02:05:22Z

Jingwen.feng001: /* Solutions: */

__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:==
Complete Notes:

https://drive.google.com/file/d/10lBjWNhVMmGosD8UCEjLrpVto9RT0top/view?usp=sharing

==Definitions:==
# Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.
# Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
# Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
# Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $ a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
# Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
# Zero (a.k.a. ''trivial'') subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
# Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
# Subring generated by a subset.
# Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

==Theorems:==
# Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
# Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
# Theorem relating fields to integral domains: Every field is an integral domain.
# Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
# Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

==Problems:==