Questions related to the algebraic structure of algebraic integers

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7
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1answer
58 views

Variant of strong approximation.

Let $K$ be a global field. Let $w$ be a place of $K$. Let $\textbf{A}^w$ be the restricted direct product over all $v$ except $w$ of the $K_v$ with respect to the subgroups $\mathcal{O}_v$. How do I ...
1
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1answer
20 views

Is $\widehat{K}L$ complete?

Let $K$ be a field and $\widehat{K}$ be a completion with respect to some valuation on $K$. Let $L$ be a finite separable extension of $K$. When regarded as a subfield of $\widehat{L}$, is ...
1
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1answer
56 views

ideal calculation and relations

Let $f$ be an integral ideal of a number field $K$ (with ring of integers $\mathcal{O}$ and let $a$ and $b$ be fractional ideals of the same. Suppose that $ab^{-1} = x\mathcal{O}$ for some $x \in K$ ...
2
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1answer
30 views

Give a basis for the ring of integers for K, a finite extension of Q in C

This question concerns the diophantine equation $$x^2+87y^2=47z^2$$ we set $K=\mathbb{Q}(\sqrt{-89})$. Define the ring of integers $o_{K}$ of K and give a $\mathbb{Z}$-basis of the form {$1,\alpha$} ...
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1answer
44 views

Number field extension

Given a number field K show that there exists a number field extension L of K such that every ideal in K becomes a principal ideal in L.
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59 views

Two algebraic number theory questions.

I have two algebraic number theory questions. See page 26 here, specifically Lemma 5.11. Lemma 5.11. The group $\mathcal{O}^\times$ is generated by roots of unity and $[\mathcal{O}^\times]^+$, the ...
2
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1answer
54 views

Equivalent condition for being a regular prime ideal

$\newcommand{\p}{\mathfrak{p}}$ $\newcommand{\tp}{\tilde{\mathfrak{p}}}$ $\newcommand{\tA}{\tilde{A}}$ I have a question about Neukirch, Algebraic Number Theory, page 92. The problem is to show the ...
1
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1answer
29 views

Topology for the decomposition of $L\otimes_K K_\frak p$

Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the ...
3
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1answer
48 views

Proving that a Prime Ideal Divides a Principal Ideal

Let $K/\mathbb Q$ be a cubic extension and $(p) = \mathfrak p_1 \mathfrak p_2 \mathfrak p_3 $ be a factorization into distinct prime ideals. Suppose that $\alpha \in \mathcal O_K$ is an integer such ...
0
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1answer
18 views

Different discriminant ideal, what are their applications?

Lots of texts online on number theory do not even mention the different ideal. Some do, but then it gets ignored and is never mentioned again. I could not find a single application for it, as if it is ...
3
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0answers
29 views

Proving unit of quartic number field is fundamental

Let $K = \mathbb Q(\alpha)$, for $\alpha$ a root of $a^4 + 4 \alpha^2 + 2 = 0$. I want to prove the group of units $\mathcal O_K^*$ equals $\langle -1, \alpha^2 + 1\rangle$. I've found the ring of ...
3
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1answer
43 views

polynomial with all roots on the unit circle

I'm wondering if the following statement is true: if all roots of a polynomial with integer coefficients are on the unit circle, then these roots are in fact roots of unity and the polynomial is a ...
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0answers
18 views

kernel of the artin map when dealing with S-ideles and S-divisors for function fields

Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group. What can be said about ...
1
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1answer
28 views

In what way is a quadratic extention to a finite field isomorphic to a finite field of higher order?

I have read (I don't remember where) that a finite field that is quadratically extended, say $\mathbb F_p[\sqrt 3]$ for example, is isomorphic to the finite field $\mathbb F_{p^2}$ (assuming the ...
1
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1answer
69 views

How many distinct roots of unity could have a sum of zero?

$\xi = \cos{\frac{ 2\pi}{n}}+i \sin{\frac{ 2\pi}{n}}$ , $i^2=-1, n$ is a positive integer. if $\xi^{a_1}+\xi^{a_2}+...+\xi^{a_k}=0$ , $a_1,a_2,...,a_k\in \{0,1,...,n-1\}$ and $a_1,a_2,...,a_k$ are ...
0
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2answers
30 views

finite extensions of discrete valuation fields: A method to find a basis

Suppose that $L|K$ is a finite extension of discrete valuation fields. Namely $w$ is a discrete valuation on $L$ extending a valuation $v$ on $K$. Now consider the respective rings of integers ...
4
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0answers
52 views

Original proof of Ljunngren's equation

The equation $$x^2=2y^4-1$$ was studied and solved by Ljunngren, who showed that 1,1 and 293,13 are the only integer solutions.However, his proof was very difficult and L.J.Mordell thought there must ...
2
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1answer
49 views

A problem about algebraic integers and roots of unity [duplicate]

Let $\lambda_1, \lambda_2,...,\lambda_n $ be roots of unity and let $z=\sum\lambda_i/n$ be an algebraic integer. To show: (a) Any conjugate of $z$ , say $z'$, is of the form $z'=\sum ...
0
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1answer
36 views

Fast way to check if finite extension is unramified?

Consider the $5$-adic quadratic extension $\mathbb{Q}_{5}(\sqrt{5})/\mathbb{Q}_{5}$. I want to check if this extension is unramified, where unramified means that the corresponding extension of ...
1
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1answer
32 views

factor ideal $(3)$ in biquadratic field

L.S., I would like to factor $(3)$ in $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of $f = X^4 + 4X^2 + 2$. I need this factoring as a part of an exercise I need to do from my course on ...
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0answers
41 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
4
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1answer
31 views

Is $\mathbb{A}_Q/\mathbb{Z}$ compact and connected?

Let $\mathbb{A}_\mathbb{Q}$be the the adéles of $\mathbb{Q}$ viewed as a group and consider $\mathbb{Z}$ as a subgroup in the natural manner. Is $\mathbb{A}_\mathbb{Q} / \mathbb{Z}$ compact and ...
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0answers
48 views

Are there practical algorithms for computing exact eigenvalues?

Numerous software implementations exist for doing diagonalization of square matrices. However, they are iterative in nature, usually based on some fixed point equation, and returns results with ...
1
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1answer
22 views

When is norm surjective mod an ideal for global fields?

Given $K/k$ Abelian, for which ideals of $\frak p\unlhd\cal O_k$ will we have $N^K_k:\cal O_K\rightarrow\cal O_k/\frak p$ surjective? $k$ is an algebraic number field. In his article "On the norm ...
2
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1answer
47 views

Equation $x^2=y^p+1$

can you help me please for solving this dophantine equation $$x^2=y^p+1$$ and if you can give me a general method to studying such equation $$x²=y^p+t$$ Thanks
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0answers
24 views

Differentiation formula in Miyake's “Modular Forms”

Miyake proves this lemma (and subsequently uses it to show the area of a hyperbolic triangle is the angle deficit): $$ (y^{-1}dz)\circ \alpha - y^{-1}dz = -2i d[log(j(\alpha,z)],$$ where $\alpha = ...
1
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1answer
45 views

prime ideals in extensions of dedekind domains

Let $A$ be a Dedekind domain and $B$ its integral closure on a finite extension of its field of fractions. Is it true that if $\mathfrak{I}$ is an ideal of $B$, then factorizing it as ...
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0answers
38 views

Methods to show that an ideal isn't principal in a quadratic number field?

Suppose $a,m\in\mathbb Z_{\ge2}$. Let's consider the ring $A=\mathbb Z[(1+\alpha)/2]$, where $\alpha^2=1-4a^m$, and the ideal $I=(a,(1+\alpha)/2)$, we need to show that $I^n$ is non-principal when ...
6
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2answers
52 views

Something wrong my proof that $\mathbb{Z}[x]$ is not a PID nor an Euclidean domain?

My proof goes as follows: Suppose for contradiction that $\mathbb{Z}[x]$ is a PID. Then the ideal generated by any irreducible element is maximal. We know that $x^2+1$ is irreducible in ...
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2answers
80 views

A question regarding independence modulo a prime ideal of a number ring

The following exercise is taken from Marcus' Number Fields and I have no clue on how to start this. I'd prefer to not be given a full answer, just a few hints on how to get started. Let $P$ be a ...
0
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1answer
29 views

Is it possible to determine what's the field in this vector space?

I was reading through a problem in number theory, and I came across the following vector space: "Let $V$ be the vector space of all functions $f:(\mathbb{Z}/N\mathbb{Z})^\times \rightarrow ...
2
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1answer
48 views

Cocycles and the first cohomology

I took algebraic number theory this semester and our professor started to teach cohomology of groups. Let $G$ be a group so that $G$ acts on an abelian group $A$. he defined $H^1(G,A)$ as the ...
5
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1answer
46 views

How do I explicitly find the norm of $I = \text{Card}(\mathbb{Z}[\sqrt{-d}]/I)?$

Say we have number field $\mathbb{Q}(\sqrt{-d})$, where $d$ is either $1$ or $2$ mod $4$, so ring of integers is $\mathbb{Z}[\sqrt{-d}]$. Suppose we have an ideal of ring of integers $I$. Now, $I$ can ...
5
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1answer
42 views

If I know all the distinct factorizations of a number, how do I use that to figure out the unique factorization of the ideal?

For example, in $\textbf{Z}[\sqrt{10}]$, we have $$6 = 2 \times 3 = (4 - \sqrt{10})(4 + \sqrt{10})$$ and $$10 = 2 \times 5 = (\sqrt{10})^2.$$ How do I use this knowledge to figure out the ...
0
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0answers
9 views

Properties of Dedekind domains considered as UFD [duplicate]

So I know that all PIDs are UFDs, but once we require a ring $R$ to be a Dedekind domain, then these two properties are equivalent, i.e. $\{\text{PID}\} = \{\text{UFD}\}$. Which condition of Dedekind ...
1
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2answers
48 views

Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
5
votes
1answer
58 views

Solving the cubic-exponential Diophantine equation $x^3+3=2^n$

The Diophantine equation $x^3+3=2^n$ has the obvious solutions $(-1,1)$,$(1,2)$ and $(5,7)$. I have been wondering if there are any other, but my attempts have been fruitless (I tried factoring it ...
0
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3answers
46 views

Quotienting out polynomial rings by polynomial-generated ideals

I'm studying in Lorenzini's Arithmetic Geometry, and it has been a while since I've taken a rigorous algebra course. I'm trying to understand a certain step in his proof that $\mathbb{Z}[\sqrt{5}]$ ...
8
votes
2answers
417 views

What information do we gain from PIDs

I am self-learning some algebraic number theory and my question is regarding the advantages to studying PIDs. I have seen that Euclidean Domains $\subseteq$ Principal Idea Domains $\subseteq$ Unique ...
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2answers
49 views

Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
3
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1answer
31 views

Cuspidal and Supercuspidal representation

Let $G$ be an algebraic group over a field $F$, and let $(\pi,V)$ be a smooth $G$-representation over an algebraic closed field $k$. Then $\pi$ is called a CUSPIDAL representation if $r(V)=0$ for any ...
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0answers
23 views

Prove the following relation regarding gamma functions:

Prove that $\forall a \in \mathbb{Z}^+, $ $\exists x, y \in \mathbb{Z}^+$ s.t. $y= \frac{(a+x)\Gamma(a-x)+(a-x) \Gamma(a+x)+ 2a}{a^2 - x^2}$ and that if $\frac{\Gamma(a+x)+1}{a+x} \notin ...
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0answers
41 views

constructing a lattice in a number field with prescribed localizations

For a prime number $p$, we denote the localization of the ring $\mathbb{Z}$ at $p$ by $\mathbb{Z}_{(p)}$. Let $k$ be an algebraic number field and denote by $\mathfrak{o}_{k}$ its ring of integers. By ...
2
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1answer
53 views

Average of sum of unit roots is an algebraic integer

Let $\alpha_1,\ldots, \alpha_n$ be roots of unity, and let $a=\frac{1}{n}\sum\alpha_i$. Then if $a$ is an algebraic integer, we have either $a=0$ or $a=\alpha_1=\dots=\alpha_n$. Why?
2
votes
2answers
45 views

Why is the group of principal units of a local field uniquely divisible by $n$?

I am reading a proof with the followings setup and claim. $K/F$ is a Galois extension of local fields with group $G$ of order $n = q^s$, where $q$ is prime and $s \geq 1$. Assume the maximal ideal ...
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1answer
35 views

How do I prove this Diophantine equation has no solutions?

Let $p$ be an odd prime number. Prove that $x^2+2y^2=pz^2$ has no solutions in natural numbers with $x, y, z$ pairwise coprime and $y$ even unless $p\equiv 1$ (mod 8). I don't understand how to ...
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0answers
13 views

Why $d^{p} =0$,and $bdd^{p-1}$=0 and $^{p-1}$ is a derivation?

I am looking example in the article:$$$$ Example:Supoose that $K$is a field of char$K=p>2$ and $L$ is its field extension $L=K(b),b\not\in K,b^{p} \in K$.Let $d$ be a $K$-linear derivation of L ...
0
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0answers
27 views

Constants in Siegel's Lemma

I've got a hopefully straightforward question to ask concerning the following lovely version of Siegel's Lemma: Let $K$ be some algebraic number field, and let $O_K$ be it's ring of integers. Define ...
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1answer
32 views

Localisation and fractional ideal

I read this in Algebraic Number Theory by A. Fröhlich & M. J. Taylor on p94: $\mathfrak o$ is a Dedekind domain with field of fraction $K$. Let $L, M$ be finitely generated torsion free ...
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1answer
39 views

$B_{\mathfrak p}$ not always a simple extension of $A_{\mathfrak p}$?

Let $B$ be the integral closure of some ring of integers $A$ in an extension of number fields, and let $\mathfrak p$ be a prime of $A$. I've seen an example where $B$ is not a simple extension of ...