0
votes
0answers
28 views

Isomorphic matrix groups over rings

I've thinking about this problem for the last couple days and I can't get anywhere. I would really appreciate some help. Is it true that, a) $\operatorname{SL}_n(\mathbb{Z}/2013\mathbb{Z})\cong ...
10
votes
2answers
221 views

Are integers mod n a unique factorization domain?

I am trying to learn abstract algebra from scratch, jolly stuff, but in the process of doing so this puzzles me: Having a ring of integers mod n, where $n=pq$ is composite, as I understand we have ...
0
votes
1answer
58 views

Show that the mod p map is a ring homomorphism

Let p be a prime and let (mod $p$)$ : Z[x] \mapsto Z_p[x]$ be the mod-p map which sends any polynomial... $f(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n \in Z[x]$ ... to the polynomial... $f(x)$(mod ...
0
votes
2answers
76 views

How to check if a function is an homomorphism?

For example: Let $$f:\mathbb{Z}_{60} \rightarrow \mathbb{Z}_{12} \times \mathbb{Z}_{20}$$ $$[x]_{60} \mapsto ([x]_{12} , [x]_{20})$$ Prove that it's well defined Check if it's a ring homomorphism ...
0
votes
2answers
41 views

Question about Divisibility

Suppose we are given the following: $p$ is a prime number; $a, c \in \mathbb{Z}$ and $ n \in \mathbb{N}$. Can I prove that there exists $m \in \mathbb{N} $ and $b \in \mathbb {Z} $ such that ...
7
votes
1answer
61 views

Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
-1
votes
1answer
85 views

Find unity of ring.

The ring {0, 2, 4, 6, 8} under addition and multiplication modulo 10 has a unity. What is that unity and how do we find it?
1
vote
2answers
48 views

Abstract Algebra: Struggles with rings

This is a part of the group of practice problem I've been working on and I'm just lost. I'm really struggling when it comes to these ring problems. Anybody who could lay out an outline for this ...
1
vote
1answer
61 views

Prove $f(x)=9x^2-5y^2-34$ has no integral roots

Prove $f(x)=9x^2-5y^2-34$ has no integral roots. I have tried working this mod 2, 3, 4, 5, and 17, and some random others, to no avail. It is for a graduate course, so I am thinking maybe I tried to ...
1
vote
0answers
248 views

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them.

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them. The above is the question, this is my attempt at an ...
4
votes
2answers
67 views

Multiplicative polinomial inverse revisited

As it usually goes with asking questions for a third person... You don't get it right the first time. Question I asked here is as follows: Let $A(x)$ be a polynomial with integer coefficients. Is ...
0
votes
2answers
408 views

Polynomials' multiplicative inverse

Let $A(x)$ be a polynomial with integer coefficients. Is there always a polynomial $B(x)$ for which $$A(x)\cdot B(x)\equiv 1\pmod n$$ (for a given integer $n$). If the answer isn't yes, an answer ...
2
votes
1answer
186 views

Application of the Chinese Remainder Theorem

Three brothers A, B and C live together and they all love eating pizza. A has the habit of eating a pizza every 5 days, B every 7 days and C every 11 days. A and C both eat pizzas together on 3 ...
5
votes
1answer
127 views

A simpler proof that if $m\mid n$ then there is a ring homomorphism from $\mathbb{Z}_n$ onto $\mathbb{Z}_m$?

Question 18.I.4 from Pinter's A Book of Abstract Algebra asks for a proof of the following, where $\mathbb{Z}_m$ and $\mathbb{Z}_n$ are treated as rings: If $n$ is a multiple of $m$, then ...
3
votes
1answer
85 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
2
votes
1answer
70 views

How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
3
votes
3answers
156 views

How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
1
vote
2answers
92 views

How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...
-1
votes
1answer
86 views

primes of type norm($a+b\sqrt{-c}$) = primes of type $1$ $mod$ $c 2^{n-1}$?

Let $a,b,c,n$ be strictly positive integers where $c$ is a prime and such that the normed ring $a+b\sqrt{-c}$ is a UFD. I noticed the primes of norm($a+b\sqrt{-1}$) are $1 \bmod 4$. Also the ...
1
vote
3answers
248 views

Question about rings and modulo multiplication tables

Why is 2=0 in K for part a)? Also I don't understand what part b) is asking you to do - what does it mean by alpha = [X], so M=etc. Could someone please explain the question and the solutions to part ...
1
vote
2answers
294 views

Finding an integer that satisfies a congruence class equation

We have integers $a = 1231940$, $b = 9935$ and $n = 3999831$ and the ring $\mathbb{Z}/n\mathbb{Z}$. Now we should find an integer $x$ that satisfies the equation $[a] \odot_n [x] = [b]$. How can such ...
1
vote
2answers
397 views

Chinese Remainder Theorem result varies

Sorry if this question is lame. First post! I was going through this book Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University In the Chapter ...
2
votes
3answers
108 views

What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$.

What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$. Are they same thing or what are significant differences between them except we can use integers greater than $n$ for ...
1
vote
2answers
257 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
2
votes
0answers
109 views

Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
1
vote
1answer
601 views

Solving a polynomial modulo an integer

Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
1
vote
2answers
483 views

Question about CRT

The question rephrased and compressed: Let $F=F_2[a]$ be a finite extension field of the field of two elements $F_2$. We are given a polynomial $R(X)\in F[X]$, and pairwise coprime irreducible ...
3
votes
3answers
272 views

Prove the sum of all numbers that do not have a multiplicative inverse mod $n$

I understand that for a number $a$ to have a multiplicative inverse in mod $n$, it must be coprime to $n$; therefore, all numbers that do not have a multiplicative inverse mod $n$ must share a factor ...