Questions tagged [number-fields]

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$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group

This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one. To calculate the Selmer group of given elliptic curve, we ...
BrauerManinobstruction's user avatar
2 votes
1 answer
252 views

On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas

I. First Set Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have, $$j=\...
Tito Piezas III's user avatar
1 vote
0 answers
104 views

The map from the decomposition group to the Galois group of the residue fields

$\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a Galois field extension. Let $L$ be a field and $K$ be a number field. Let $B$ be a valuation subring of $L$ and let $A$ be the preimage of $B$ in $K$ (i.e ...
Michail Karatarakis's user avatar
1 vote
1 answer
94 views

The map from the ring of integers to the residue field of a valuation subring is surjective

Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...
Michail Karatarakis's user avatar
1 vote
0 answers
26 views

Natural Density of Norms of ideals in a given ideal class

Some time ago, Landau proved the following formula for general number fields: $I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below ...
George Bentley's user avatar
3 votes
3 answers
409 views

Irreducibility of polynomials over some number fields

Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions). Now, in order to ...
Jean's user avatar
  • 467
5 votes
1 answer
220 views

Relation between $G_{\mathbb{Q}_p}$ for different primes

Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known. It is well known that this group embeds ...
kindasorta's user avatar
0 votes
0 answers
39 views

Can one $n$-sect a general angle using a "rods and hinges" construction?

This question was inspired by a post by Alon Amit. It is a standard algebra result that it is impossible to $n$-sect a given angle using only a ruler and compass. In fact, it is impossible to trisect ...
Yonah Borns-Weil's user avatar
3 votes
0 answers
107 views

Embedding number fields in fields with class number prime to $p$

Let $p$ be a fixed prime. Question: For any number field $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that the class number of $L$ is prime to $p$? Moreover, for any ...
stupid boy's user avatar
4 votes
1 answer
478 views

Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$

For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative. I believe that the following statement is true: $$\zeta_K\...
Permutator's user avatar
1 vote
0 answers
67 views

Liouville numbers with some "special" convergents

Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which $$ 0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
Jean's user avatar
  • 467
4 votes
1 answer
202 views

Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be ...
Nic Banks's user avatar
0 votes
0 answers
115 views

Is there a number field $K$ such that $K^\times / \mathbb{Q}^\times$ is finitely generated? [duplicate]

I think I have a simple proof that the only fields with finitely generated multiplicative groups are finite. What about if we take $K$ a number field and mod out $\mathbb{Q}^\times$ from its ...
Bma's user avatar
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0 votes
1 answer
179 views

Class number of imaginary quadratic fields

Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a ...
user492144's user avatar
1 vote
1 answer
84 views

Size of sets associated to Gaussian integers

Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$ containing all solutions of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,...
Roland Bacher's user avatar
-1 votes
1 answer
149 views

Set of all primes $p$ that split in $\mathbb{Q}\left(\sqrt{-k}\right)$

Let $k$ be a squarefree positive integer. We know that a prime $p$ splits in $K=\mathbb{Q}(\sqrt{-k})$ if and only if $-k$ is a quadratic residue mod $p$. My question is: can we explicitly determine ...
user491084's user avatar
9 votes
1 answer
635 views

Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

This is a cross-post! For the original post on SE (9 upvotes, no answer) see: https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
Luvath's user avatar
  • 145
5 votes
2 answers
268 views

Additivity of Elliptic Curve Rank over Compositum of Fields

Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
Jeff H's user avatar
  • 1,342
6 votes
1 answer
444 views

Algorithm for computing whether a cubic field is monogenic?

I am interested in existing algorithms to compute whether a given non-cyclic, non-pure cubic extension $K/\mathbb{Q}$ is monogenic or not, and if so, to give me a defining polynomial for the integral ...
edward cornfoot's user avatar
9 votes
1 answer
388 views

Square root in number field

I'm trying implement an algorithm that, for an element $b$ of a number field $\mathbb{Q}(\alpha)$, if it is a square in $\mathbb{Q}(\alpha)$ (i.e., $\exists x\in\mathbb{Q}(\alpha):x^2=b$), computes ...
Tippisum's user avatar
  • 131
6 votes
2 answers
293 views

Algebraic numbers which prescribed degree which does not belong to some fields

In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
Jean's user avatar
  • 467
11 votes
1 answer
464 views

Sign and coefficients of fundamental unit of quadratic field

Is there any way to determine whether the fundamental unit of a quadratic field has negative or positive norm, except by actually computing the unit to all of its (many) digits? And, similarly, ...
Michael Beeson's user avatar
5 votes
2 answers
135 views

Dihedral extension unramified at primes dividing order of group?

Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
user404920's user avatar
1 vote
0 answers
68 views

The localization map for the Mordell-Weil group of elliptic curves over finite Galois extensions

Let $L/K$ be a finite Galois extensions of number fields and $E/K$ be an elliptic curve. Denote by $\mathcal{F}$ the localization map \begin{equation} \mathcal{F}: H^1(G,E(L)) \rightarrow \bigoplus_{v ...
A. Maarefparvar's user avatar
1 vote
0 answers
76 views

How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$

Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...
matt stokes's user avatar
2 votes
0 answers
90 views

Number fields with given discriminant

In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
Joe Shipman's user avatar
3 votes
0 answers
121 views

Congruence of elements implies congruence of norms for central simple algebras

I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...
Radu T's user avatar
  • 757
4 votes
1 answer
185 views

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$

I want to examine nontrivial examples of what we call Iwasawa class formula, $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only ...
BrauerManinobstruction's user avatar
4 votes
1 answer
320 views

Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
Sky's user avatar
  • 693
3 votes
0 answers
166 views

Decomposition of primes in cyclotomic extensions and their ramifications

Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$. So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...
user100603's user avatar
4 votes
0 answers
107 views

Factorization in the ring of integers of a particular biquadratic number field, and questions about norms

Consider the number field $K={\mathbb Q}[\sqrt{2},\sqrt{3}]$ and its ring of integers ${\mathcal O}_K$. I have been doing some calculations with this number field as a toy example, to see what can be ...
Yemon Choi's user avatar
  • 24.8k
2 votes
0 answers
128 views

Dirichlet unit theorem for finite rings

Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements ...
Denis Marcinkov's user avatar
3 votes
0 answers
99 views

Furtwängler's family of irreducible polynomials

In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form: $$ p(x) = x^4 \prod_{i=1}^{...
wandersam's user avatar
  • 125
1 vote
0 answers
96 views

Translation of a paper by Dedekind (an integral basis for pure cubic fields)

I am studying Introductory Algebraic Number Theory written by S. Alaca and K. Williams. The authors mention the theorem concerning an integral basis for pure cubic fields but do not provide proof. ...
Infinity_hunter's user avatar
9 votes
1 answer
557 views

Is it true that this ideal must be principal? (proof verification)

Let $L/K$ be a (abelian, Galois) quadratic extension of number fields with $\text{Gal}(L/K)$ generated by $\sigma$ and $\mathfrak{p} = \alpha\mathcal{O}_K$ a principal prime ideal of $\mathcal{O}_K$. ...
wyoumans's user avatar
  • 287
2 votes
1 answer
147 views

Does a quaternion algebra exist over a number field that is split over some infinite real places, but not others?

Let $F$ be a totally real number field having at least two different real embeddings $\sigma_1 : F \hookrightarrow \mathbb{R}$ and $\sigma_2 : F \hookrightarrow \mathbb{R}$. Does a quaternion algebra $...
Doryan Temmerman's user avatar
3 votes
0 answers
371 views

Fundamental Units in $\mathbb{Z}[\sqrt{d}]$ with $d \equiv 1 \mod 4$

It is well known and often cited how the fundamental units for the number ring of $\mathbb{Q}[\sqrt{d}]$ look like. In the case of $d\equiv 2,3 \mod 4$ the number ring is $\mathbb{Z}[\sqrt{d}]$ and in ...
Thomas Lachmann's user avatar
6 votes
1 answer
327 views

Galois module theory: from global to local

Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \...
Lios's user avatar
  • 213
4 votes
1 answer
191 views

Norm groups of number fields

I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
Melanka's user avatar
  • 549
2 votes
0 answers
94 views

Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$

I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
LoneStar's user avatar
  • 153
2 votes
0 answers
139 views

Has there been much research on the Iwasawa theory of bi-quadratic fields?

The simplest non-trivial example of $\mathbb{Z}_p$-extensions happen over imaginary quadratic fields and I've seen a good amount of research done here (from my understanding still a lot to learn). I ...
matt stokes's user avatar
14 votes
1 answer
1k views

How Dirichlet proved Dirichlet's unit theorem for general number fields?

For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r_1+r_2-1$. It seems that standard algebraic ...
J.Li's user avatar
  • 751
1 vote
0 answers
105 views

What conditions on primes makes it non principal?

Let $K$ be a number field and $\mathfrak{p}$ is prime of $K$ what condition I can have on $\mathfrak{p}$ so it becomes non-principal ideal?
user11333's user avatar
  • 343
0 votes
0 answers
90 views

How to determine if a unramifed prime split or not?

Let $K$ be the Number field and $L$ be finite extension where $\mathfrak{p}$ prime of K is unramified. Are there any conditions on $\mathfrak{p}$ so that I can say $\mathfrak{p}$ splits completely in ...
SUNIL PASUPULATI's user avatar
1 vote
0 answers
87 views

A subgroup of the $n$-Selmer group

Let $p$ be an odd prime and for the purpose of this question let $n$ be an integer which is a power of $p$. Let $E$ be an elliptic curve over a number field $F$. The $n$-Selmer group, denoted by $S_n(...
debanjana's user avatar
  • 1,089
4 votes
2 answers
250 views

Richaud-Degert type quadratic extensions

A Richaud-Degert type real quadratic field is a number field of the form $K = \mathbb{Q}(\sqrt{d})$ where $d = {(an)}^2 + ka > 0$ for positive integers $a, n$ and $k \in \{ \pm 1, \pm 2, \pm 4 \}$, ...
Melanka's user avatar
  • 549
1 vote
0 answers
81 views

Exponents of powers in cyclotomic extensions

For every number field $K$ and for every $a \in K$ let $$\epsilon(K;a) := \sup\!\left\{e \in \mathbb{N} : \exists b \in K \text{ such that } a = b^e\right\}. $$ Now let $K$ be a number field and let $...
O_k's user avatar
  • 11
8 votes
1 answer
239 views

What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
Keshav Srinivasan's user avatar
5 votes
1 answer
170 views

On the stabilizer in $\mathrm{GL}(2,\mathbb{Z})$ of a real quadratic irrationality

$\DeclareMathOperator\GL{GL}$Let $\theta$ be a real quadratic irrationality with discriminant $\Delta$; let $\mathcal{O}_\Delta$ denote the resulting quadratic order of discriminant $\Delta$ in $\...
Branimir Ćaćić's user avatar
3 votes
2 answers
403 views

Correspondence between binary quadratic representations and proper ideals of quadratic number fields

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$Fix $d < 0$, a fundamental quadratic discriminant and $n$ a positive integer. Suppose $Q$ is a primitive binary quadratic form of ...
Melanka's user avatar
  • 549

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