Tagged Questions

3
votes
1answer
34 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
55 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
119 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
71 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
57 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
99 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
101 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
157 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
76 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
115 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
79 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
251 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
455 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
214 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 ...