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Math 361, Spring 2017, Assignment 10

2017-05-10T21:03:17Z

Anamaria.Ronayne: /* Questions */

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

# Relative algebraic closure (of a field $F$ in an extension $E$).
# $\overline{\mathbb{Q}}$ (a.k.a. the ''field of algebraic numbers'').
# Algebraically closed field.
# Prime subfield (of a field).
# Characteristic (of a field).
# $GF(p^n)$.

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

# Theorem characterizing algebraically closed fields (i.e. giving several conditions equivalent to the property of being algebraically closed).
# Fundamental Theorem of Algebra.
# Theorem concerning degrees of irreducible polynomials in $\mathbb{C}[x]$.
# Theorem concerning degrees of irreducible polynomials in $\mathbb{R}[x]$.
# Theorem restricting the cardinality of a finite field.
# Theorem concerning existence of finite fields of certain cardinalities.
# Theorem relating finite fields of equal cardinality.

==Carefully describe the following procedures:==

# Procedure to factor polynomials in $\mathbb{C}[x]$.
# Procedure to factor polynomials in $\mathbb{R}[x]$.
# Procedure to construct finite fields.

==Solve the following problems:==

# Section 33, problems 1, 2, and 3.
# Construct a field with $125$ elements. (That is, describe how to write down elements of your field, give rules for addition and multiplication together with examples of these calculations, and explain why all this results in a field. You do ''not'' need to write down complete addition or multiplication tables.)
# Factor the polynomial $x^3 - 1$ over $\mathbb{C}$. ''(Hint: the roots all lie on the unit circle in the complex plane, and they are equally spaced around it. See the Wikipedia article on "Roots of unity" for more information.)''
# Factor the polynomial $x^3 - 1$ over $\mathbb{R}$.
# Show that $\overline{\mathbb{Q}}$ is algebraically closed, as follows: let $p=a_0+\dots+a_nx^n$ be any non-constant polynomial with coefficients in $\overline{\mathbb{Q}}$.
::(a) Explain why $p$ must have a root in $\mathbb{C}$. Choose such a root, and call it $\beta$.
::(b) Using the Dimension Theorem, show that $\mathbb{Q}(a_0,\dots,a_n)$ (i.e. the smallest subfield containing $\mathbb{Q}$ and the coefficients of $p$) must be finite-dimensional over $\mathbb{Q}$.
::(c) Using the classification of simple extensions, show that $\mathbb{Q}(a_0,\dots,a_n,\beta)$ is finite-dimensional over $\mathbb{Q}(a_0,\dots,a_n)$.
::(d) Using the Dimension Theorem a second time, conclude that $\beta$ is algebraic over $\mathbb{Q}$, and hence lies in $\overline{\mathbb{Q}}$.

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

==Questions==
1)'''Theorem characterizing algebraically closed fields (i.e. giving several conditions equivalent to the property of being algebraically closed):'''1)Every non-constant is in F[x] has a root. 2)Every non-constant in f[x] factors as a product of linear factors ("splits") 4) Every irreducible in F[x] has a degree 1. 5)Every algbrica extension of F is trivial.

2)'''Fundamental Theorem of Algebra:''' $\mathbb{C}$ is algebraically closed $x^5 +x^4+\pi(x^3)+ (e+i)x^2 +2x)$ has roots in $\mathbb{C}$

3)'''Theorem concerning degrees of irreducible polynomials in $\mathbb{C}[x]$:''' ??? Could not find at all so sad.

4)'''Theorem concerning degrees of irreducible polynomials in $\mathbb{R}[x]$:''' Any irr element $\mathbb{R}[x]$ has degree 1 or 2.

5)'''Theorem restricting the cardinality of a finite field:''' Suppose $F$ is a finite field then, $|F|$is a power of a prime.

6)'''Theorem concerning existence of finite fields of certain cardinalities:'''?? sort of idea. For every prime power $p^n$ there is a field $F$ with $|F|$=p.

7)'''Theorem relating finite fields of equal cardinality''': Any two finite filed with the same cardinality are in fact isomorphic.

My question is are my theorems correct? But more important I could not find theorem 3 and not sure completely on theorem 6. Can you make sure the others are correct as well.

==Solutions==

Math 361, Spring 2017, Assignment 11

2017-04-15T18:51:00Z

Anamaria.Ronayne: /* Questions */

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

# Compass-and-straightedge construction.
# Constructible number.

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

# Theorem concerning sums, differences, products, and inverses of constructible numbers.
# Theorem concerning square roots of constructible numbers.
# Theorem concerning the degrees of constructible numbers over $\mathbb{Q}$.
# Theorem concerning duplication of the cube.
# Theorem concerning squaring the circle.

==Describe the following compass-and-straightedge constructions:==

# Perpendicular bisector.
# Dropping a perpendicular.
# Constructing a parallel.
# Multiplication of lengths.
# Inversion of length.
# Square root of length.

==Solve the following problems:==

# Determine whether the following numbers are constructible:
:: (a) $\frac{\sqrt{5}-1}{4}$.
:: (b) $\sqrt[6]{7}$.
:: (c) $\alpha$ where $\alpha$ is a root of the polynomial $x^3+3x-12$. ''(Hint: use Eisenstein's criterion.)''

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

==Questions==
A) $\frac{\sqrt(5) - 1}{4}$

$\alpha = \frac{\sqrt(5) - 1}{4}$

$4\alpha = \sqrt(5) - 1$

$4\alpha +1 = \sqrt(5)$

$(4\alpha +1)^2 = \sqrt(5)^2$

$ 16\alpha^2 + 8\alpha + 1= 5$

$ 16\alpha^2 + 8\alpha + 4= 0$

Therefore because the problem is of degree two it is constructible.
Is my reasoning totally off?

The Eisenstein's criterion: Can be applied like this:

$8x^3+6x^2-9x+24$ where $p=3$ because $3^2=9$ and does not go into 24 evenly therefore the equation is irreducibility.

C)$x^3+3x-12$ where $p=3$ because $3^2=9$ and does not go into 12 evenly therefore the equation is irreducibility.
How does this help us determine weather the following is constructible or not?

==Solutions==

Math 361, Spring 2017, Assignment 6

2017-04-10T13:18:59Z

Anamaria.Ronayne: /* Questions */

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

# Princpial ideal (generated by an element $a$ in a commutative ring $R$).
# Principal ideal domain (a.k.a. PID).
# Standard representative (of a coset in the quotient ring $F[x]/\langle m\rangle$ where $F$ is a field and $m$ is a polynomial with coefficients in $F$).
# Prime ideal.
# Maximal ideal.

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

# Fundamental theorem of ring homomorphisms.
# Classification of ideals in $F[x]$ ("Every ideal of $F[x]$ is...").
# Theorem relating prime ideals to integral domains.
# Theorem relating maximal ideals to fields.

==Solve the following problems:==

# Section 27, problems 1, 2, 3, 10, 11, 15, 16, and 17.

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

==Questions==
I having trouble with Section 27, problem 16).
Not quite sure where to start. The question is find a prime ideal of $\mathbb{Z} \times \mathbb{Z}$ that is not maximal. I was under the assumption that every prime ideal was also maximal.
In problem 1 of Section 27 say find all the prime ideals and all maximal ideals of $\mathbb{Z}_6$
so...the work is...
$ 1 = \mathbb{Z_6}$

$(2) = \{0,2,4\}$

$(3) =\{ 0,3\}$

$(6)= \{0\}$

$(2) = \{0,2,4\}$
Are both prime so they are both prime and maximal.
$(3) =\{ 0,3\}$

and
$(6)= \{0\}$ is neither prime nor maximal.

==Solutions==

Math 361, Spring 2017, Assignment 9

2017-04-09T01:39:05Z

Anamaria.Ronayne: /* Questions */

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

# Vector space (over a field $F$).
# Subspace (of a vector space).
# Subspace generated (a.k.a. ''spanned'') by a set $S$.
# Linear relation (in a set $S$).
# Linearly independent.
# Basis.
# $\mathrm{dim}_F V$ (a.k.a. the ''dimension of $V$ over $F$'').
# $[E:F]$ (a.k.a. the ''degree of $E$, regarded as an extension of $F$'').

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

# Theorem concerning the existence of bases.
# Theorem relating the cardinalities of two bases for the same vector space.
# Theorem relating bases to unique expansion of vectors as linear combinations.
# Dimension Formula.
# Corollary of the Dimension Formula analogous to Lagrange's Theorem in group theory.
# Corollary concerning subextensions of field extensions of prime degree.
# Theorem characterizing algebraic elements in terms of finite-dimensional subextensions.
# Theorem concerning sums, products, and inverses of algebraic elements.

==Solve the following problems:==

# Section 30, problems 1, 4, 5, 6, 7, and 8.
# Section 31, problems 1, 3, 5, 7, 22, 23, 24, 25, and 29.

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

==Questions==
The first homework question talks about bases for $\mathbb{R}^2$ no two of which have a vector in common. I am not sure where to start to find the three bases. Is the next several question ask a given basis for the indicated vector space over the field. Not sure where to start, with those questions. Where do I start to solve the problems?
::Any two vectors in $\mathbb{R}^2$, neither of which is a scalar multiple of the other, will form a basis for $\mathbb{R}^2$. So, for example, $\begin{bmatrix}1\\1\end{bmatrix}$ and $\begin{bmatrix}1\\2\end{bmatrix}$ will form a basis.
::Any simple algebraic extension has a "standard basis" $\{1,\alpha,\dots,\alpha^{d-1}\}$ where $\alpha$ is the generator and $d$ is the degree of $\alpha$ over the base field. So, for example, $\mathbb{Q}(\sqrt[5]{2})$ has $\mathbb{Q}$-basis $\{1, \sqrt[5]{2}, (\sqrt[5]{2})^2, (\sqrt[5]{2})^3, (\sqrt[5]{2})^4\}$. -[[User:Steven.Jackson|Steven.Jackson]] ([[User talk:Steven.Jackson|talk]]) 10:28, 4 April 2017 (EDT)
I have a problem with question 22 section 31. The question is: Let $(a+bi) \in \mathbb{C}$ where $a, b \in \mathbb{R}$ and $b \neq 0.$ Show that $\mathbb{C} =\mathbb{R}(a+bi)$.(22 is not in the back of the book as an answer) Thanks again for the help.

==Solutions==

Math 361, Spring 2017, Assignment 9

2017-04-09T01:37:55Z

Anamaria.Ronayne: /* Questions */

Math 361, Spring 2017, Assignment 7

2017-04-06T00:34:50Z

Anamaria.Ronayne: /* Questions */

__NOTOC__

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

# List of prime ideals of $\mathbb{Z}$.
# List of maximal ideals of $\mathbb{Z}$.
# List of prime ideals of $F[x]$.
# List of maximal ideals of $F[x]$.
# Theorem characterizing when the quotient ring $F[x]/\langle m\rangle$ will be a field.

==Solve the following problems:==

# Section 27, problems 5, 7, and 9.
# Construct a field with exactly eight elements. Make addition and multiplication tables for your field. Is your field isomorphic to $\mathbb{Z}_8$? ''(Hint: first find an irreducible cubic in $\mathbb{Z}_2[x]$.)''

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

==Questions==
List of prime ideals of $\mathbb{Z}$-If R is a ring with unity 1, then the map $\phi : \mathbb{Z} \rightarrow R$ given by $ \phi(n) =n \cdot 1$ for $n \in \mathbb{Z}$ into $R$.

List of maximal ideals of $\mathbb{Z}$: Could not fine at all, in my notes or book?

List of prime ideals of F[x]:Could not fine at all, in my notes or book?

List of maximal ideals of F[x]: Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a field.

Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field: Let p(x) be an irreducible polynomial in F[x]. If p(x) divides $r(x)s(x)$ for $r(x)$, $s(x) \in F[x]$, then either p(x) divides r(x) or p(x) divides s(x).

'''So let recap my question here is are the theorems that I listed here are they correct? As well as can you tell me where in the book I might find the, List of maximal ideals of Z and the List of prime ideals of F[x]:
I would greatly appreciate the help.'''

==Solutions==

Math 361, Spring 2017, Assignment 7

2017-04-06T00:33:40Z

Anamaria.Ronayne: /* Questions */

__NOTOC__

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

# List of prime ideals of $\mathbb{Z}$.
# List of maximal ideals of $\mathbb{Z}$.
# List of prime ideals of $F[x]$.
# List of maximal ideals of $F[x]$.
# Theorem characterizing when the quotient ring $F[x]/\langle m\rangle$ will be a field.

==Solve the following problems:==

# Section 27, problems 5, 7, and 9.
# Construct a field with exactly eight elements. Make addition and multiplication tables for your field. Is your field isomorphic to $\mathbb{Z}_8$? ''(Hint: first find an irreducible cubic in $\mathbb{Z}_2[x]$.)''

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

==Questions==
List of prime ideals of Z-If R is a ring with unity 1, then the map $\phi : \mathbb{Z} \rightarrow R$ given by $ \phi(n) =n \cdot 1$ for $n \in \mathbb{Z}$ into $R$.

List of maximal ideals of Z: Could not fine at all, in my notes or book?

List of prime ideals of F[x]:Could not fine at all, in my notes or book?

List of maximal ideals of F[x]: Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a field.

Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field: Let p(x) be an irreducible polynomial in F[x]. If p(x) divides $r(x)s(x)$ for $r(x)$, $s(x) \in F[x]$, then either p(x) divides r(x) or p(x) divides s(x).

'''So let recap my question here is are the theorems that I listed here are they correct? As well as can you tell me where in the book I might find the, List of maximal ideals of Z and the List of prime ideals of F[x]:
I would greatly appreciate the help.'''

==Solutions==

Math 361, Spring 2017, Assignment 7

2017-04-05T17:01:32Z

Anamaria.Ronayne: /* Questions */

Math 361, Spring 2017, Assignment 9

2017-04-02T01:28:59Z

Anamaria.Ronayne: /* Questions */

Math 361, Spring 2017, Assignment 9

2017-04-01T23:49:20Z

Anamaria.Ronayne: /* Questions */

Math 361, Spring 2017, Assignment 4

2017-02-18T17:51:59Z

Anamaria.Ronayne: /* Questions */

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

# Divisibility relation (in an integral domain).
# Associate relation (in an integral domain).
# Irreducible element (of an integral domain).
# Unique factorization domain.

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

# Universal mapping property of $R[x]$ (this is not stated concisely in the book; it is the statement concerning "generalized evaluation homomorphisms" that we gave in class).
# Theorem concerning polynomial long division.
# Fundamental theorem of arithmetic.
# Theorem concerning unique factorization of polynomials.
# Factor theorem.
# Bound on the number of roots of a polynomial.

==Solve the following problems:==

# Section 23, problems 1, 3, 9, 11, 13, and 27.
# Working in $\mathbb{Z}_5[x]$, find all associates of the polynomial $x^2+3$.

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

==Questions==
I am don't Understand how the follow picture below works I believe in understanding the picture I can Understand question 3 on section 23 pg. 218
which asks about $f(x)=x^5-2x^4+3x-5$ and $g(x)=2x+1$ in $\mathbb{Z}_{11}$

[[File:math 361 homework.jpg]]

==Solutions==

File:Math 361 homework.jpg

2017-02-18T17:49:25Z

Anamaria.Ronayne: Anamaria.Ronayne uploaded a new version of File:Math 361 homework.jpg

File:Math 361 homework.jpg

2017-02-18T17:48:22Z

Anamaria.Ronayne:

Math 361, Spring 2017, Assignment 4

2017-02-18T17:47:55Z

Anamaria.Ronayne: /* Questions */

File:Math 361.jpg

2017-02-18T17:37:38Z

Anamaria.Ronayne:

Math 361, Spring 2017, Assignment 4

2017-02-18T17:36:12Z

Anamaria.Ronayne: /* Questions */

Math 141, Spring 2016, Assignment 1

2017-02-18T17:27:45Z

Anamaria.Ronayne: /* Solve the following problems: */

__NOTOC__
''The beginner...should not be discouraged if...he finds that he does not have the prerequisites for reading the prerequisites.''

: - P. Halmos

==State the following formulas (you do not need to prove them):==

# Formula for $\int\frac{1}{\sqrt{1-x^2}}\,dx$.
# Formula for $\int\frac{1}{1+x^2}\,dx$.
# Formula for $\int\tan(x)\,dx$.
# Substitution rule.
# Integration by parts.

==Solve the following problems:==

# Section 2.7, problems 15, 17, 19, 21, and 22.
# Section 6.1, problems 3, 7, 11, 35, 41, 43, 45, 59, and 66.
# Copy Theorem 35 on page 193, volume 1. (Really. Copy it. This will help you to remember the formulas - and eventually you will need to remember them.)
# Copy Theorem 45 on page 263.
# Choose an appropriate domain restriction for the secant function, use this to define an inverse secant function, then find the derivative of the inverse secant and convert this to an integration formula. (Contrary to what I said in class, this ''does'' result in a new integration formula, one version of which appears as the third formula in Theorem 46 on page 267.)

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

==Solutions:==

File:Math 361 question 3.jpg

2017-02-18T17:25:57Z

Anamaria.Ronayne:

Math 141, Spring 2016, Assignment 1

2017-02-18T17:24:46Z

Anamaria.Ronayne: /* Questions: */

Math 361, Spring 2017, Assignment 3

2017-02-13T20:36:46Z

Anamaria.Ronayne: /* Solve the following problems: */

__NOTOC__

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

# Polynomial function defined by a polynomial expression.
# Zero (a.k.a. ''root'') of a polynomial.
# Degree (of a polynomial).
# $R[x, y]$.
# $R[x_1,\dots,x_n]$.

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

# Bound on the degree of the sum of two polynomials.
# Bound on the degree of the product of two polynomials.
# Exact formula for the degree of the product of two polynomials, in case the base ring is an integral domain.

==Solve the following problems:==

# Section 22, problems 5, 7, 9, 11, 13, and 15.
# Prove that if $D$ is an integral domain, then so is D[x]. ''(Hint: the only hard part is showing that the product of two non-zero polynomials is non-zero. So suppose $f,g\neq0$. What can you say about $\mathrm{deg}(f)$, $\mathrm{deg}(g)$, and $\mathrm{deg}(fg)$? Alternatively, what can you say about the highest-degree term of $fg$?)''
# Let $D$ be any integral domain. The ''field of rational expressions in one indeterminate, with coefficients in $D$'' is the field $D(x) = \mathrm{Frac}(D[x])$. Write down two or three sample elements of $\mathbb{Q}(x)$, then show how to add, multiply, and invert them.
# More generally, the ''field of rational expressions in $n$ indeterminates'' is the field $D(x_1,\dots,x_n) = \mathrm{Frac(D[x_1,\dots,x_n])}$. Give sample elements and sample calculations, as above, for the field $\mathbb{Q}(x,y)$.
# Let $p$ be any prime number. Show that $\mathbb{Z}_p(x)$ is always an infinite field (even though $\mathbb{Z}_p$ itself is a finite field).

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

==Questions:== I have no idea where to start with Question 5 in section 22. '''How many polynomials are there of degree $\leq 3 \in \mathbb{Z}_2[x]$? include (0)''' Any suggestions I have been looking at this problem for a few hours now still have no clue.

Solutions: Question 11) $\phi_4(3x^{106}+5x^{99}+2x^{53})$ in $\mathbb{Z}_7$

1) First use Fermat's theorem. $a^{p-1}= 1(mod$ $p$) in our case $a^{7-1}=1 (mod$ $7$) So $a^6 = 1 (mod$ $7$)

2)Now take each power in the polynomial function $(3x^{106}+5x^{99}+2x^{53})$ So you get $106=6 \cdot 17+4$, $99=6 \cdot 16 + 3$, $53=6 \cdot 8 +5$

3) Now plug in 4 into the new x and the new exponent powers. so that you get $3 \cdot 4^4(4^6)^{17} + 5 \cdot 4^3(4^6)^{16} + 2 \cdot 4 ^5 \cdot (4^6)^8$

$3 \cdot 4^4(1)^{17} + 5 \cdot 4^3(1)^{16} + 2 \cdot 4 ^5 \cdot (1)^8$

$3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$

$3 \cdot 256 + 5 \cdot 64 + 2 \cdot 1024$

$ 768+320+2048$

$3136$

$\frac{3136}{7}$=448 Remainder of 0

thus the answer is zero

My question for this problem is there something I am missing that would make less step. Specify is their a way to make this line $3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$ be simple without doing out by hand or without a calculator?

Math 361, Spring 2017, Assignment 3

2017-02-13T20:34:27Z

Anamaria.Ronayne: /* Solutions: */

__NOTOC__

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

# Polynomial function defined by a polynomial expression.
# Zero (a.k.a. ''root'') of a polynomial.
# Degree (of a polynomial).
# $R[x, y]$.
# $R[x_1,\dots,x_n]$.

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

# Bound on the degree of the sum of two polynomials.
# Bound on the degree of the product of two polynomials.
# Exact formula for the degree of the product of two polynomials, in case the base ring is an integral domain.

==Solve the following problems:==

# Section 22, problems 5, 7, 9, 11, 13, and 15.
# Prove that if $D$ is an integral domain, then so is D[x]. ''(Hint: the only hard part is showing that the product of two non-zero polynomials is non-zero. So suppose $f,g\neq0$. What can you say about $\mathrm{deg}(f)$, $\mathrm{deg}(g)$, and $\mathrm{deg}(fg)$? Alternatively, what can you say about the highest-degree term of $fg$?)''
# Let $D$ be any integral domain. The ''field of rational expressions in one indeterminate, with coefficients in $D$'' is the field $D(x) = \mathrm{Frac}(D[x])$. Write down two or three sample elements of $\mathbb{Q}(x)$, then show how to add, multiply, and invert them.
# More generally, the ''field of rational expressions in $n$ indeterminates'' is the field $D(x_1,\dots,x_n) = \mathrm{Frac(D[x_1,\dots,x_n])}$. Give sample elements and sample calculations, as above, for the field $\mathbb{Q}(x,y)$.
# Let $p$ be any prime number. Show that $\mathbb{Z}_p(x)$ is always an infinite field (even though $\mathbb{Z}_p$ itself is a finite field).

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

==Questions:== I have no idea where to start with Question 5 in section 22. '''How many polynomials are there of degree $\leq 3 \in \mathbb{Z}_2[x]$? include (0)'''

Solutions: Question 11) $\phi_4(3x^{106}+5x^{99}+2x^{53})$ in $\mathbb{Z}_7$

1) First use Fermat's theorem. $a^{p-1}= 1(mod$ $p$) in our case $a^{7-1}=1 (mod$ $7$) So $a^6 = 1 (mod$ $7$)

2)Now take each power in the polynomial function $(3x^{106}+5x^{99}+2x^{53})$ So you get $106=6 \cdot 17+4$, $99=6 \cdot 16 + 3$, $53=6 \cdot 8 +5$

3) Now plug in 4 into the new x and the new exponent powers. so that you get $3 \cdot 4^4(4^6)^{17} + 5 \cdot 4^3(4^6)^{16} + 2 \cdot 4 ^5 \cdot (4^6)^8$

$3 \cdot 4^4(1)^{17} + 5 \cdot 4^3(1)^{16} + 2 \cdot 4 ^5 \cdot (1)^8$

$3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$

$3 \cdot 256 + 5 \cdot 64 + 2 \cdot 1024$

$ 768+320+2048$

$3136$

$\frac{3136}{7}$=448 Remainder of 0

thus the answer is zero

My question for this problem is there something I am missing that would make less step. Specify is their a way to make this line $3 \cdot 4^4 + 5 \cdot 4^3 + 2 \cdot 4 ^5$ be simple without doing out by hand or without a calculator?

Math 361, Spring 2017, Assignment 3

2017-02-13T19:56:02Z

Anamaria.Ronayne: /* Questions: */

Math 361, Spring 2017, Assignment 2

2017-02-05T00:31:18Z

Anamaria.Ronayne: /* Questions: */

__NOTOC__

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

# Formal fraction.
# Equivalence (of formal fractions).
# Fraction.
# $\mathrm{Frac}(D)$.
# Polynomial function.
# Polynomial expression.

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

# Universal mapping property of $\mathrm{Frac}(D)$.

==Solve the following problems:==

# Section 21, problems 1 and 2. (In both problems, you are being asked to use the universal mapping property to find a "concrete model" of the field of fractions, as we did in class.)
# Section 22, problems 1 and 3.
# Prove Euler's theorem. ''(Hint: Since $\mathrm{gcd}(a,n)=1$, we can regard $a$ as an element of the group of units $G(\mathbb{Z}_n)$. The order of this group is $\phi(n)$. Now see Theorem 10.12 on page 101 of the text.)''

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

==Questions:==
I am having trouble with describing the field F of quotients of the integral subdomain $D=\{n+mi|n,m \in \mathbb{Z}\}$ of complex numbers. Describe means give the elements of the complex numbers that make up the field of quotients of D in the complex numbers. (The elements of D are the Gaussian integers.)
The answer in the back of the book says $\{q_1 + q_2i|q_1, q_2 \in \mathbb{Q}\}$ I want to understand how to get to the answer what are the key steps

==Solutions:==

Math 361, Spring 2017, Assignment 2

2017-02-05T00:30:24Z

Anamaria.Ronayne: /* Questions: */