Questions related to the algebraic structure of algebraic integers

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Minimal polynomials and degree of field extension

I have a cyclotomic field $\mathbb{Q}(\zeta_3)$, and want to know how I can find a minimal polynomial of $\zeta_{10}$, and $\zeta_{12}$. I have determined that both the polynomials should be of ...
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Cyclotomic polynomial in $\mathbb{Z}/(p)$

Let $p$ be prime, $n \in \mathbb{N}$ and $p \nmid n$. $\Phi_n$ is the $n$-th cyclotomic polynomial. How can I find the maximum $n \in \mathbb{N}$ (with $p \nmid n)$ so that $\Phi_n$ splits into ...
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density theorems and class field theory

am I correct in thinking that the frobenius density theorem (it says that the Dirichlet density of the set of primes of K that split completely in an extension L is 1/[L:k]) is sort of one of the main ...
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$ K(\sqrt{a})$ is unramified if and only if $a \mid d_K$ and $a \equiv 1 \mod{4}$.

Let $K$ be an imaginary quadratic field of discriminant $d_K$ and let $K(\sqrt{a})$ be a quadratic extension where $a \in \mathbb{Z}$. Then $K \subset K(\sqrt{a})$ is unramified if and only if $a$ can ...
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Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
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Divisibility in the ring of integers.

For example, let $R=\Bbb Z [\sqrt{-5}]$, and I want to explain $3$ does not divide $2-\sqrt{-5}$. I think the following proof will be right: Suppose $3(a+b\sqrt{-5})=2-\sqrt{-5}$, then taking ...
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Is there any concept similar to unique factorization that applies to exponential operators?

We can talk about prime numbers over multiplication but is there any similar concept that applies to exponential operators or other hyperpowers like tetration? Can we use what we know about UFDs to ...
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Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
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51 views

Minkowski's theorem for non-symmetric convex bodies

Minkowski's theorem for convex bodies states that every convex, symmetric subset of $\mathbb{R}^d$ whose volume is larger than $2^d$ contains a non-zero integer point. All the proofs I've seen rely on ...
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1answer
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Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
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Are there number fields based on ultraradicals?

So you can create quadratic fields, cubic fields, and quartic fields by just taking the nth root of some integer, and some are even unique factorization domains or principle ideal domains, like $\Bbb ...
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Are there unique factorizations for weyl algebras?

I just read about Weyl algebras, and they sound like neat little toys that are similar in a number of ways to polynomials. However its curious to me that they are non-commutative, and I was wondering ...
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4answers
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Number theory - rational number

Are there any $x, y$ that fit in below $\sqrt{4y^2-3x^2}$ such that an rational number is yielded. Appreciate if explanation is given.
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Prime number of ${\bf Z}$ and prime element of ${\bf Z}[i]$

I am looking at the class note from graduate number theory: Let $p$ be prime number in ${\bf Z}$ and r be prime element in ${\bf Z}[i]$. If $r$ is an associate of $p$, then $p$ is congruent to $3$ ...
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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 ...
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1answer
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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 ...
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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 ...
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2answers
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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 ...
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$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ 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 ...
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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? ...
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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 $2^n$.
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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 $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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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?
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 = ...
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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 ...
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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 ...
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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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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 $O_{K}=o_{k}[A]$.

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 ...
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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. ...
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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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Quadratic reciprocity problem

How can I use quadratic reciprocity to prove that $-3$ is a quadratic residue $\pmod p$ if and only if $p=2$ or $p \equiv 1 \pmod 6$ and deduce that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_p ...
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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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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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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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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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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 ...