3
votes
0answers
28 views

Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
-1
votes
3answers
36 views

Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
4
votes
2answers
70 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
0
votes
2answers
47 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
-1
votes
0answers
22 views

A commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$

Can anyone give an example of a commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$ and $\forall x \,\, x \cdot x = 0$? ...
0
votes
1answer
40 views

Can a ring be both commutative and anticommutative?

As the title says, can a ring ever be both commutative and anti-commutative? By anti-commutivity, $a \cdot b+b \cdot a =0$. Edit: A ring must have infinitely many numbers.
2
votes
1answer
41 views

Prove that for $P(X) \in \mathbb{Z}[X]$ the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\}$ is infinite

Prove that for $P(X) \in \mathbb{Z}[X]$, $P(x)$ non-constant, the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\}$ is infinite. could someone please give ...
1
vote
1answer
30 views

Valuation associated to a non-zero prime ideal of the ring of integers

I have a question from Frohlich & Taylor's book 'Algebraic Number Theory', p.64. I will keep the notation used there. Let $K$ be a number field, $\mathcal o$ its ring of integers. Let $\mathfrak ...
4
votes
2answers
98 views

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! :) ...
1
vote
1answer
56 views

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 ...
2
votes
1answer
101 views

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, ...
1
vote
1answer
47 views

$(2,1+\sqrt{-5})$ 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 ...
0
votes
1answer
47 views

$\Bbb Z[i\alpha]$ 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 ...
1
vote
1answer
30 views

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 ...
0
votes
1answer
19 views

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 $c|b$ ...
1
vote
0answers
69 views

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 ...
4
votes
0answers
67 views

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 ...
0
votes
2answers
67 views

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?
0
votes
2answers
52 views

Show that 2 is a prime in the ring $Z[\frac{-1+\sqrt{-3}}2]$

My progress: Let's take $a\in Z[\frac{-1+\sqrt{-3}}2]$ such that $a|2$, and function $l(x)=x \bar x$. $a|2$ $\Rightarrow$ $2=a*b$ $\Rightarrow$ $l(a*b)=l(a)l(b)=4=l(2)$ If $z \in ...
3
votes
2answers
132 views

Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...
2
votes
1answer
136 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if ...
3
votes
1answer
106 views

Find two elements that don't have a gcd in a subring of Gaussian integers

Find two elements in the domain $R := \{ x + 2y \sqrt {-1} \mid x,y \in \mathbb{Z} \}$ that do not have a gcd. I have no idea how to start. But I know if we consider $R^\prime = \{ x + y \sqrt ...
2
votes
0answers
75 views

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 $\mathbf{Q}(\sqrt{-d})$ ...
0
votes
2answers
39 views

$f(x)\in D[x]$ 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 ...
3
votes
1answer
66 views

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 ...
4
votes
3answers
163 views

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 ...
0
votes
0answers
47 views

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 ...
1
vote
3answers
34 views

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 ...
0
votes
0answers
70 views

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 ...
2
votes
1answer
62 views

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 ...
1
vote
1answer
35 views

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 ...
4
votes
3answers
310 views

$1^n +2^n + \cdots +(p-1)^n \mod p =$?

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 ...
2
votes
2answers
211 views

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 ...
2
votes
1answer
380 views

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 $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then ...
4
votes
1answer
99 views

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 ...
0
votes
2answers
61 views

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 ...
2
votes
0answers
62 views

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 ...
4
votes
1answer
91 views

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 $\mathbb{Z}\left[\sqrt{5}\right]$?
4
votes
4answers
273 views

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 ...
3
votes
1answer
83 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, ...
0
votes
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 ...
4
votes
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) = ...
0
votes
1answer
114 views

Smallest Subring

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 $ST$ ...
0
votes
2answers
97 views

Zero Divisor Rings

Let $R$ be a ring, and let $a,b \in R$ such that $ab \ne 0$. Show that $ab$ is a zero divisor if and only if $a$ is a zero divisor or $b$ is a zero divisor. I understand that a zero divisor is $ab = ...
2
votes
4answers
129 views

Show that if $c_1 + c_2\sqrt{5}$ divides $n$ in ${\bf{O}}[\sqrt{5}]$, then so does $c_1 - c_2\sqrt{5}$

I have a ring: $${\bf{O}}[\sqrt{5}] = \{c_1 + c_2\sqrt{5}: (c_1 \in \mathbb{Z} \wedge c_2 \in \mathbb{Z}) \lor (c_1 + \frac{1}{2} \in \mathbb{Z} \wedge c_2 + \frac{1}{2} \in \mathbb{Z}) \}.$$ I ...
2
votes
1answer
80 views

Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
0
votes
0answers
77 views

Probability that two Gaussian integers are divisible

Let $z = x+ iy \in \mathbb{Z}[i]$ and let $a+ib \in \mathbb{Z}[i]$ with $a^2 + b^2 \equiv 1 \mod{4}$. What is the probability that $a+ib$ divides $x + iy$ in $\mathbb{Z}[i]$? This question would ...
3
votes
2answers
151 views

Probability that $x \equiv 3 \pmod{4}$

I am working on a number theory project and I am interested in the following statement: What is the probability that an integer $x$ has the property that $|x| \equiv 3 \mod{4}$? This seems ...
3
votes
4answers
147 views

Non-commutative or commutative ring or subring with $x^2 = 0$

Does there exist a non-commutative or commutative ring or subring $R$ with $x \cdot x = 0$ where $0$ is the zero element of $R$, $\cdot$ is multiplication secondary binary operation, and $x$ is not ...
9
votes
2answers
375 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...