Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

0
votes
1answer
31 views

Prime number of Z and prime element of Z[i]

I am looking at the class note from graduate number theory: Let p be prime number in Z and r be prime element in Z[i]. If r is an associate of p, then p congruent 3 (mod 4). I spent hours trying to ...
0
votes
0answers
20 views

Methods for computing subextensions for a n-th cyclotomic field.

So the problem is 1)find all quadratic and cubic subextensions of $\mathbb{Q}[\zeta^{527}]$ and 2)describe how it's primes split completely in the cubic subextensions. Can you give me some ...
1
vote
0answers
38 views

Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
0
votes
0answers
15 views

Discriminant of p-adic $\mathbb{Q}_{p}[\phi]$, where $0=f(\phi)=\phi^{p}-\phi-1$

Any suggestions using the minimal polynomial? How about $D_{\mathbb{Q}_{p}[\phi]/\mathbb{Q}_{p}}=(-1)N_{\mathbb{Q}_{p}[\phi],\mathbb{Q}_{p}}(f')$? But foremost I prefer you suggest me a correct ...
0
votes
0answers
29 views

Finding number of roots of a polynomial in p-adic integers $\mathbb{Z}_{p}$

The problem is to find the number roots of $x^3+25x^2+x-9 $ in $\mathbb{Z}_{p}$ for p=2,3,5,7. I read this equivalent to having a root mod $p^{k}~\forall k\geq 1$. By Newton's lemma I can get whether ...
0
votes
1answer
26 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$

Let $n$ be a square-free integer such that $n\equiv 0,2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
0
votes
1answer
17 views

Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
3
votes
1answer
48 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
3
votes
2answers
40 views

Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
0
votes
0answers
20 views

Integer Solutions of eliptic curve

I neet to find all integer solutions to the equation $2x^2+25=y^3$ Can you please help me?
1
vote
1answer
18 views

What does “A mod P generates the residue class field extension” mean?

We have K and finite algebraic extension L. P is a prime ideal in $O_{L}$ over prime $p\in O_{K}$ and $A\in O_{L}$. Then the problem says $\bar{A}:=A ~mod ~P$ generates the residue class field ...
1
vote
1answer
35 views

Geometric meaning of being integrally closed in some overring

The geometric counterpart of integrally closed rings (in their fraction fields) are normal varieties, as described in this MathOverflow post. Is their a similar notion in algebraic geometry for being ...
0
votes
0answers
10 views

ANT Frohlich Proposition 3, (v). Induced map of dual modules has the same determinant

$R$ is a Dedekind domain, $V$ is an $n$ dimensional vector space over its quotient field $K$, $B(-,-)$ is a $K$-bilinear form on $V$, and $M, N \subseteq V$ are free $R$-modules of rank $n$. Also ...
0
votes
0answers
11 views

Can the General Number Field Sieve be used to factor in any unique factorization domain?

Related slightly to my question about factoring in quadratic rings, can you use the general number field sieve to factor in any unique factorization domain? Can you use it in any UFD that isn't the ...
0
votes
1answer
12 views

Uniqueness of unramified extensions of $\mathbb{Q}_{p}$

So I showed that $\mathbb{Q}_{p}[\theta]$ is an unramified extension of degree p, where $0=g(\theta)=\theta^{p}-\theta-1$. But it also follows that $\mathbb{Q}_{p}[\phi]$ is an unramified extension ...
3
votes
1answer
29 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
2
votes
3answers
27 views

Ramification and inertia degree for $\mathbb{Q}_{p}[a]$ where $0=g(a)=a^{3}+25a^{2}+a-9$

The problem is to find e and f for p-adic rationals for p=2,3,5,7. Because g is not Eisenstein for each p, the extension will not be tottaly ramified and thus $3=ef\Rightarrow e=1$ and $f=3$. I feel I ...
0
votes
1answer
17 views

multiplying ideals

At the moment I am learning algebraic number theory and I have seemed to be missing some basic understanding. How do you multiply ideals For example $$(2,1+\sqrt{-5})^2 = ...
0
votes
1answer
22 views

Galois group of $L/\mathbb{Q}$ is generated by inertia groups

L is a Galois extension and we want Galois group G to be generated by the inertial groups for all primes Any suggestions? Here is my proof (in process): The intermediate field $L_{I}$ for the ...
3
votes
1answer
63 views

What are the elements of $(\mathbb{Q}_{2}/(\mathbb{Q}_{2})^{2})^{\times}$?

The answer is $\{-1, \pm 2,\pm 5, \pm 10\}$ and I can't even figure out why 10 is there. I mean the 2-adic unit rationals. It turns out that ...
1
vote
1answer
27 views

What are the roots of unity quadratic integers?

The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi ...
0
votes
1answer
41 views

Questions on a proof of “All prime ideals of a Dedekind domain are invertible”

I tried to prove this theorem : All prime ideals of a Dedekind domain is invertible. i.e, For every prime ideal $\mathfrak{p}$ of Dedekind domain $R$, there exists $\mathfrak{p}^{-1} \subseteq ...
0
votes
0answers
32 views

If $|\Phi(A)|=|\Pi|$, then $O_{K}=o_{k}[A]$.

Here is the problem: K/k is a finite algebraic extension, $\Pi$ is a prime element in K, $A\in O_{K}$, p,P is the maximal ideal of k and K. We have that $\bar{A}:=A ~mod ~P$ generates the residue ...
1
vote
0answers
33 views

Show $L:=\mathbb{Q}_{2}[\sqrt{1+\sqrt{-2}}]$ is completely ramified.

This equivalent to $f(x)=x^{4}-2x^2 +3$ being Eiseinstein. We have $|1|_{2}=1$, $|2|_{2}=2^{-1}$ and $|3|=|1|$. So 3 is a 2-adic unit. Thus, f is not Eisenstein and so L is not completely ramified. ...
3
votes
0answers
30 views

primes splitting completely in cyclic extensions

Let $K$ be a quadratic number field. It is a well known result that a prime $p$ splits completely in $K$ if and only if $\left(\frac{d_K}{p}\right)=1$. What about cubic extensions? Can we find ...
9
votes
2answers
245 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
1
vote
0answers
35 views

Generalizations of results on the sum of divisors function over $\mathbb{Q}$ to number fields

Consider the sum of divisor function $$ \sigma_0(n) = \sum_{d\mid n} 1. $$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the ...
1
vote
2answers
46 views

Proof about Number Fields

It is a known result that if $\alpha$ is an algebraic integer in a number field $K$, i.e. $\alpha \in \mathcal{O}_K$, then its trace and norm are integers. I am looking over a proof of this, which ...
0
votes
1answer
23 views

Relating Galois groups related via completions

Let $K$ be a number field, and let $K_\mathfrak p$ denote the completion of $K$ at the prime $\mathfrak p$ of $K$. I'm wondering what can be said (if anything useful) that relates the Galois groups ...
1
vote
1answer
49 views

Roots of unity in a residue field of a Cyclotomic extension

Neukirch makes the following assertion in Algebraic Number Theory: Let $L = \mathbb{Q}(\zeta)$ where $\zeta$ is a primitve $n$th root of unity. Let $p$ be an integer coprime to $n$. For any prime ...
2
votes
1answer
54 views

Lower bound on divisors of $\Phi_n(n) $

Take the nth cyclotomic polynomial $\Phi_n(x)$ and let $\phi$ be the Euler totient function. I can prove that all divisors $d$ of $\Phi_n(n)$ are such that $d \ge \phi(n)$ or $d = 1$. The proof is ...
2
votes
0answers
21 views

Recovering congruence conditions from the Hilbert class polynomial for idoneal numbers

Before I can ask my question, I need to introduce some terminology and background. Statement 1: Let $n$ be one of Euler's 65 convenient numbers. Then we can find congruence conditions such that ...
1
vote
1answer
49 views

$[K(a):F(a)]=[K:F]$ if $a$ is transcendental over $K$.

Let $F$ and $K$ be subfields of the complex number $\mathbb{C}$ such that $K$ is a finite extension of $F$. Let $a\in \mathbb{C}$. If $a$ is not algebraic over $K$, prove that $[K(a):F(a)]=[K:F]$. I ...
-1
votes
0answers
19 views

Showing that a set of units form a multiplicutive group

Let $R$ be a ring and let $U(R)$ be the set of units in $R$. Show that $U(R)$ is a multiplicative group.
1
vote
2answers
36 views

Methods for finding Number of roots in $\mathbb{Q}_{p}$ for a polynomial

The problem is: find how many roots in $Q_{p}$ does $x^{3}+25x^{2}+x-9$ have for p=2,3,5,7. I found for $\mathbb{Z}_{p}$. Is there a way to extend from here? Also, can you suggest me a book or link ...
4
votes
2answers
51 views

General methods for solving p-adic inequality $|a^2+b|_{p}<p^{-k}$ for $a\in \mathbb{Z}$

The problem is: find integer a that satisfies the 5-adic norm inequality $|a^2+6|_{5}<5^{-4}$. I tried in vain finding it computationaly. Are there any methods from number theory to help me solve ...
2
votes
0answers
27 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
2
votes
0answers
57 views

If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
0
votes
0answers
27 views

How is the Dirichlet unit theorem effectively used to solve diophantine equations

Can someone give me an example or a link to one of its use to solve a diophantine equation ? More precisely : this theorem explains the structure of the units of the ring of integers of a number ...
0
votes
2answers
38 views

number cubic polynomials possible

Let $p(x)$ be a cubic polynomial with integral coefficients , such that $p(a)=b$, $p(b)=c$, $p(c)=a$ for $a,b,c$ being distinct integers . find number of such possible polynomials.
4
votes
2answers
63 views

Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
2
votes
1answer
34 views

Is the ring of integers of a local field an open subgroup?

I apologize if my question is a bit naive, but I don't have much experience in number theory and sometimes get very confused. Suppose $K$ is a non-archimedean local field (essentially a completion of ...
1
vote
1answer
55 views

Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...
1
vote
1answer
21 views

Find a basis for $B/\mathfrak{B}^e$

In the context of algebraic integers, I would like to solve te following problem. Let $A \subset B$ be two rings, $\mathfrak{p}$ a prime ideal of $A$ and $\mathfrak{B}$ a prime ideal of $B$ lying ...
5
votes
1answer
145 views

The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
2
votes
1answer
54 views

Existence of finite morphism to projective line

As we all know, Belyi's theorem says: A complex curve $X$ is defined over a number field, if and only if there exists a finite morphism $t:X\to \mathbb{P}^1_\mathbb{C}$ of varieties over $\mathbb{C}$ ...
0
votes
1answer
21 views

Annihilation and localization

Can somebody help proofing the following lemma. Let $x$ be an element of a module $M$, and let $\mathfrak{a}$ be its annihilator. Let $\mathfrak{p}$ be a prime ideal of $A$. Then $(Ax)_{\mathfrak{p}} ...
1
vote
1answer
26 views

Integral closures and Galois extensions

I was reading in Lang's Algebraic Number Theory (Second Edition page 15-16) and the following proposition occured. Proposition 14. Let $A$ be integrally closed in its quotient field $K$, and let $B$ ...
2
votes
2answers
60 views

A ring automorphism in cyclotomic field

My friend asked me a question, see this. I've thought about that for some time, but I cannot do it, I don't want to let her wait too long, can you explain it for me? Thanks in advance!
0
votes
1answer
22 views

Prove that GCD domain is integrally closed

In the case $R$ is a UFD, the proof of that $R$ is integrally closed essentially uses a key fact: If gcd($a,b)=1$, then gcd$(a^n,b)=1$. The proof of this relies on UFD property: if $a,b$ are ...