# Tagged Questions

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### The prime elements of the ring $\mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right]$.

I have to study the prime elements of the ring $\mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right]$. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
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### Number of solutions of some congruence equations.

How many $[u]\in(\mathbf{Z}/ab\mathbf{Z})^\ast$ satisfy the equations $u\equiv 1 \bmod \ a$, $u\equiv 1 \bmod \ b$? I somehow believe that the answer might be $(a,b)$. Is this actually true? Is the ...
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### Euclid's Lemma for Euclidean Ring.

Question: If $R$ is a euclidean ring and $\pi\in R$ is irreducible, prove that $\pi\mid\alpha\beta$ implies $\pi\mid\alpha$ or $\pi\mid\beta$. A solution is to prove all euclidean rings are PIDs, ...
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### $(2,1+\sqrt{-5})$ has integral basis $2$, $1+\sqrt{-5}$

$2,1+\sqrt{-5}$ is an integral basis for the ideal generated by them in $\mathbb{Z}[\sqrt{-5}]$. Is there a quick way to see this? What if these two are replaced with another pair? My method: Write ...
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### $\Bbb Z[i\alpha]$ UFD's

I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask ...
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### Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
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### Definition of h.c.f./g.c.d. not fitting with $\mathbb{Z}$

In my lecture notes, and also on many websites, the definition of the highest common factor of two elements in an integral domain $R$, say $a$ and $b$, is an element $c$ such that: $c|a$ and $c|b$ ...
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### Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
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### Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
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### Division in ring $Z[\frac{-1+\sqrt{-3}}2]$

Let $w=\frac{-1+\sqrt{-3}}2$,Find $q,r \in Z[w]$ such that $3+5w=(2-w)q+r$ What's the best way to approach this kind of questions?
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### General Primality Conditions in the UFD $\mathbf{Q}(\sqrt{-d})$

Suppose $\mathcal{O}_{\mathbf{Q}\left(\sqrt{-d}\right)}$ is a UFD, so $d=1,2,3,7,11,19,43,67,163$. Are there general criteria determining whether an element in the integers of $\mathbf{Q}(\sqrt{-d})$ ...
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### $f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$.

Let $D$ be a principal ideal domain and $F$ be its quotient field. Prove that $f(x)\in D[x]$ is irreducible if and only if $f(x)$ is irreducible over $F[x]$. I only obtained the proof for ...
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### Examples of Dedekind rings with infinite class number

I am looking for explicit examples of Dedekind rings with infinite class number. In most books on algebraic number theory there is a standard example (before or after proving that the class number is ...
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### similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
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### Peculiarities of an extended integer ring $\mathbb{Z}[i C]$

For an extended integer ring consisting of $\mathbb{Z}[i C] = \{ x + iC y \mid x,y \in \mathbb{Z} \}$, here $C$ is a real constant (I guess it being complex would change nothing), what real numbers ...
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### Let $m$ and $n$ be integers in the ring of integers. Show that if $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$

Hello everyone working on the problem in the title...it's an if and only if proof so two directions to show start with the fact $n\mathbb Z$ is a subset of $m\mathbb Z$ and want to show $n=mx$ for ...
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### System of congruences that do not satisfy CRT assumptions (via algorithm)

Let $x_i,a_i\!\in\!\mathbb{Z}$. The following procedure solves a system of congruences $$x \equiv x_i\pmod{a_i}\;\;\text{ for }i\!=\!1,\ldots,n$$ when $a_i$ are pairwise coprime. Assume that ...
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### Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?

I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
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### A certain ideal of a valuation ring

This is a question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
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### $1^n +2^n + \cdots +(p-1)^n \mod p =$?

Calculate for every positive integer $n$ and for every prime $p$ the expression $$1^n +2^n + \cdots +(p-1)^n \mod p$$ I need your help for this. I don't know what to do, but I'll show you what I ...
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### Factoring 1001 in $\Bbb Z[\sqrt 7]$

I am solving the problem of factoring 1001 into prime elements in $\Bbb Z[\sqrt 7]$. I have a couple of questions regarding this. It seems that $\Bbb Z[\sqrt 7]$ is an Euclidean domain. But I do ...
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### What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$

So far, I know if $p$ is a rational prime, then $(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$. $(2)$ If $p\equiv1\mod4$ then $p=Ï€_1 Ï€_2$ where $Ï€_1$ and $Ï€_2$ are conjugate, Then ...
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### Ring of integers of a degree $5$ extension

Consider the polynomial $P(X) = X^5 - X + 1 \in \mathbb{Q}[X]$, and let $x \in \mathbb{C}$ be a root of $P(X)$. Let $K = \mathbb{Q}(x)$. How can you prove that the ring of integers $\mathcal{O}_K$ is ...
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### The action of a Galois group on a prime ideal of a Dedekind domain

This is a slight variant of a question I asked earlier. Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let ...
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### The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If ...
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### A question on factorial rings

Is 31 irreducible in the ring $\mathbb{Z}\left[\sqrt{5}\right]=\left\{a+b\sqrt{5}:a,b\in\mathbb{Z}\right\}$ ? And is it prime in $\mathbb{Z}\left[\sqrt{5}\right]$?
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### The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$

I'm really confused with this one... How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality? Does ...
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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 ${\scr{B}}=\{a_1+b, a_2+b, ... 1answer 57 views ### Ring Isomorphism Proof Let$p$be a prime with$p \equiv 1 (\mod 4 )$. I am trying to show that$\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$is a ring isomorphism. I am not really sure how to ... 2answers 105 views ### Direct product of polynomial rings Let$n = pq$, where$p$and$q$are distinct primes. I am trying to show that: $$\mathbb{Z}_n[X] \cong \mathbb{Z}_p[X] \times \mathbb{Z}_q[X].$$ Would it suffice to say that$\rho(np) = ...
Suppose that $S$ and $T$ are subrings of a ring $R$. Show that their ring-theoretic product $ST$ is a subring of $R$ that contains $S \cup T$, and is the smallest such subring. I understand that $ST$ ...