Questions related to the algebraic structure of algebraic integers

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How find representatives of $(\mathbb{Z}/2\mathbb{Z})^{3}$ and why $\mathbb{Z}_{2}^{*}/(\mathbb{Z}_{2}^{*})^{2}\simeq (\mathbb{Z}/8\mathbb{Z})^{*}$

I guess 1, 2, 5 and 10 because 5 is a cubic non-residue of 2. Right? Also, why $\mathbb{Z}_{2}^{*}/(\mathbb{Z}_{2}^{*})^{2}\simeq (\mathbb{Z}/8\mathbb{Z})^{*}$? One direction is from Hensel's lemma ...
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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 ...
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If $a$ is a root for $f(x):=5x^3-7x^2+3x+6$, then find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$

The problem is: If $a\in \mathbb{Z}_{7}$ is a root for $f(x):=5x^3-7x^2+3x+6$ s.t. $|a-1|_{7}<1/7$, then find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$. By the way, I found such root ...
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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 ...
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If $|\Phi(A)|=|\Pi|$, then $O_{L}=o_{k}[A]$.

Here the problem: K/k is a finite algebraic extension, $\Pi$ is a prime element in K, $A\in O_{K}$, p is the maximal ideal of k and P the one for K. We have that $\bar{A}:=A mod P$ generates the ...
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Is $L:=\mathbb{Q}_{2}[\sqrt{1+\sqrt{-2}}]$ Galois extension of $\mathbb{Q}_{2}$? completely ramified?

I think it is because normal: The minimal polynomial is $f(x)=(x^{2}-1)^{2}+2$ which has roots $1,\sqrt{-2}, \sqrt{1+\sqrt{-2}},\sqrt{-2}\sqrt{1+\sqrt{-2}}$ and those are all contained in L. ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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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$[K(a):F(a)]=[K:F]$ 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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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 ...
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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 ...
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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 ...
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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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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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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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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
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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
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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
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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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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 ...
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1answer
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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
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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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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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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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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 ...
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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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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 ...
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1answer
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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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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
28 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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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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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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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 ...
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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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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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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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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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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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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 $z = ...
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$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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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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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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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 ...