Questions related to the algebraic structure of algebraic integers

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2
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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 ...
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1answer
33 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 ...
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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.
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2answers
25 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
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2answers
38 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
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0answers
23 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
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0answers
51 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 ...
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0answers
25 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 ...
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2answers
37 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.
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2answers
57 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 ...
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1answer
32 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 ...
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1answer
53 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 ...
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1answer
19 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 ...
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1answer
140 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
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1answer
50 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 $\mathbb{C}$ ...
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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}} ...
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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 $B$ ...
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2answers
58 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!
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1answer
20 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 ...
2
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2answers
62 views

What does $conclude$ mean in this sentence?

My friend asked me a question, but I don't know the meaning of the sentence Conclude that $\sigma_n$ is a ring automorphism here, does it mean Prove that $\sigma_n$ is a ring automorphism or Make the ...
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0answers
61 views

Is my proof good enough?

Prove: The product of any three consecutive natural numbers is divisible by 6. 6|n (n+1) (n+2). If n is an odd number (n=(2 t+1), t is any natural number), then (n+1)= ((2 t+1)+1) an even number. So ...
2
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1answer
50 views

Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
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1answer
54 views

ramification index divides $q-1$ (cyclotomic fields)

Let $K$ be an abelian extension of $\mathbb{Q}$ with $[K:\mathbb{Q}] = p^m$. Suppose $q$ is a prime $\neq p$ which is ramified in $K$. Let $Q$ be a prime of $K$ lying over $q$. Prove that $e(Q|q)$ ...
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1answer
27 views

Factorization into prime ideals

I citate Serge Lang's Algebra (Second edition, page 23). If $A$ is a discrete valuation ring, then in particular, $A$ is a principal ideal ring, and any finitely generated torsion-free module $M$ ...
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0answers
46 views

Can we realize cyclic groups as class groups of number fields?

I was referred to this question in a previous post. I was wondering if something is known about the stronger condition, i.e. when $G$ is cyclic. Some heuristics: Every dihedral group $G ...
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0answers
13 views

How to calculate efficiently number of pairs?

How can we calculate efficiently number of pairs (a,b) such that a*b<=n where n ranges from 1 to n-1?
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0answers
45 views

Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong ...
6
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2answers
41 views

Number theory problem - contradiction

In an algebraic proof (for my problem it doesn't matter which proof) I have a special setting: $a,b,c \in \mathbb{Z}, \text{gcd}(a,c)=1,b<c \ \text{and} \ a \in \left\lbrace 1, \ldots , ...
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1answer
39 views

Greatest common divisor theorem proof

I wanted to prove an algebraic theorem and therefore I would need a statement like the following: In a commutative Ring $R$ it states for $a,b,c \in R$ with $a \neq b, a \leq c, b \leq c$ that ...
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0answers
31 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
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0answers
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If prime P of K is inert in all intermediate fields but L, then Galois group is cyclic of prime order.

So inert means $r=e(Q|P)=1$ so there is a unique prime over prime P in all intermediate fields but L (right?). I already proved that this implies G is cyclic of prime order. But the hint says to use ...
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0answers
10 views

Inertial group is trivial and decomposition is the whole Galois group if the prime is inert

I don't see how that fact follows from $[L_{E}:L_{D}]=f(Q|P)=[L:K]=|G|$ for P inert and Q lying over it. So I want to show $[G:D]=[L_{D}:K]=1$ and $[G:E]=[L_{E}:K]=[L:K]=|G|$. One way would be ...
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2answers
22 views

The way conjugation acts on embeddings

Let $K$ be a field, $F$ be complex conjugation, and let $K_{\mathbb{C}} = \prod_{\tau} \mathbb{C}$, where $\tau$ is taken over the set of embedding $K \hookrightarrow \mathbb{C}$. Then for $z = ...
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0answers
16 views

$d_{P}\mid d_{Q}f(Q\mid P)$ where $d_{P}$ is the order of the class containing prime ideal $P$.

$P$,$Q$ are prime ideals in number fields and $f(Q\mid P)$ is the residual degree of $Q$ over $P$. Here is an attempt: I already showed $f:[I]\to [N_{L/K}(I)]$ is a homomorphism. We have ...
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1answer
48 views

Class number of $\Bbb Q(a)$ with $a^3-a+1 = 0$ is $1$

Show that class number of $\Bbb Q(a)$ with $a^3-a+1 = 0$ is $1$. Any hints? Suggested theorems? I don't want a solution. Mistaken Attempt: Apply Kummer-Dedekind to $x^3-x+1 \pmod p$. For just ...
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0answers
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r a primitive root of unity show, (r-1)/((r^k) - 1) is an algebraic integer in Q(r)

I left out some hypotheses in the title to keep things short, so here is the full form: Let r be a primitive mth root of unity for m>1 and let k be a positive integer such that gcd(m,k)=1. Show (r-1) ...
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0answers
60 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 ...
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1answer
19 views

Residue class field of ring of integers is finite

Let $\mathcal{o}_K$ be a ring of algebraic integers. I have a proof for the fact that $\mathcal{o}_K$ is a free module of finite rank over $\mathbb{Z}$. Now, let $\mathcal{p}$ be a prime ideal of ...
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0answers
23 views

Finding a particular integral basis of the cyclotomic field

Let $\zeta_{39}$ be a primitive $39$th root of unity. How can I prove that all the conjugates of $\zeta_{39}$ form an integral basis of $\mathbb{Q}(\zeta_{39})$? This is from the paper "Cyclotomic ...
3
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1answer
71 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
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0answers
44 views

Find extension of $\mathbb{Q}$ containing components of eigenvectors of a matrix

Given a matrix $\mathbf{A} \in \mathbb{Z}^{d \times d}$ I need to find an algebraic number $a$ of minimal degree, such that all eigenvalues and eigenvector's coordinates of $\mathbf{A}$ belong to the ...
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1answer
32 views

Factorising ideals in $\mathbb{Z}[\sqrt{10}]$

I understand how to factorize ideals into prime ideals when they are of the form $(p)$, by Dedekind's Theorem, but I can't factorize ideals like $(4+\sqrt{10})$ in $\mathbb{Z}[\sqrt{10}]$. I can ...
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4answers
70 views

Ordering the solutions to Pell's Equation

Let $S$ be the set of positive integer pairs $(x,y)$ such that $x^2 - d y^2 = -4$ or $x^2 - d y^2 = 4$, where $d$ is fixed as the discriminant of a real quadratic number field. I'm trying to show ...
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1answer
63 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
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2answers
90 views

Solved Problems in Algebraic Number Theory

I'm not too sure whether this is the right place to ask this (and please correct me if it is not), but I'm currently studying a course in Algebraic Number Theory and would like to be pointed in the ...
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0answers
107 views

What are going to change of our view if $\pi+e$ is a rational? [closed]

It is well known that there's no conclusion now whether $\pi+e$ is a rational or not. Just for curiosity, what will happen if we know the answer?
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1answer
60 views

$\mathbb{Q}(\sqrt[p]{q}) \neq \mathbb{Q}(\sqrt[p]{r})$ for $p,q,r$ primes and $q \neq r$.

Let $p,q$ and $r$ be primes in $\mathbb{Z}$ with $q \neq r$. Let $\sqrt[p]{q}$ denote any root of $x^p-q$ and let $\sqrt[p]{r}$ denote any root of $x^p - r$. I need to prove that ...
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2answers
57 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
5
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1answer
36 views

Abelian extensions under inclusion, and their conductors

Suppose $K$ is a number field, and let $L$ and $L'$ be two abelian extensions of $K$, with conductors $C(L/K)=\mathcal{C}$ and $C(L'/K)=\mathcal{C}'$, respectively. Question: Is it true that the ...
4
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1answer
93 views

Finding a primitive element for the Hilbert Class Field of $\mathbb{Q}(\sqrt{-14})$

I am trying to solve exercise 6.18 in the book of David Cox. I have tried to provide as much context as possible to make the situation as clear as possible for the reader. I have solved ...