Questions tagged [number-fields]

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Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Question: Let $p$ be an ...
Ramin's user avatar
  • 1,362
18 votes
1 answer
1k views

A set of generators for $\bar{\mathbb{Q}}$

Two questions: Does there exist a sequence $\alpha_1,\alpha_2,...$ of algebraic numbers with degrees $d_1,d_2,...$ s.t. for each $i$, $d_i|d_{i+1}$ and $\alpha_i= p_i(\alpha_{i+1})$ with $p_i$ a ...
Mostafa's user avatar
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12 votes
0 answers
418 views

Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
BernyPiffaro's user avatar
11 votes
0 answers
438 views

A congruence involving roots of unity

Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity. Computational evidence ...
user94267's user avatar
  • 305
9 votes
0 answers
252 views

How small may the discriminant of an $S_d$-field be?

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
Vesselin Dimitrov's user avatar
8 votes
0 answers
173 views

The density of minimal polynomials

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. By the primitive element theorem there exists $\alpha \in K$ such that $K = \mathbb{Q}(\alpha)$. Let $$\displaystyle S_K = \{\...
Stanley Yao Xiao's user avatar
8 votes
0 answers
188 views

The Stickelberger's annihilation theorem over an arbitrary number field

Let $p$ be a prime and let $C = \mathbb{F}_p^\times$. Then Gal$(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, where $\zeta_p$ is a primitive $p$th root of unity, may be identified with $C$ in the obvious way. Let ...
Cindy Tsang's user avatar
6 votes
0 answers
496 views

Genus of a number field

I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as $$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
Leonardo Lanciano's user avatar
6 votes
0 answers
681 views

What are the fastest ways to calculate class number of number fields?

Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$? I am aware that the question is broad but any argument would be helpful. Some basic approaches I know:...
Ninja's user avatar
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5 votes
0 answers
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free subgroups of $SL_2(\mathbb{Z[i]})$

The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc} 1 & 2\\ 0 &...
Ofir's user avatar
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5 votes
0 answers
706 views

Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...
352506's user avatar
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5 votes
0 answers
538 views

Elliptic curve with no points in a number field

The following is probably well-known (I'd appreciate a link): for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 −...
Albertas's user avatar
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4 votes
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123 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
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4 votes
0 answers
275 views

Analytic class number formula for orders

In the article "The analytic class number formula for orders in products of number fields" (https://arxiv.org/pdf/1604.04564.pdf), it is shown that the analytic class number formula holds for ...
pencil_sharpener's user avatar
3 votes
0 answers
49 views

On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
Reyx_0's user avatar
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3 votes
0 answers
121 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
3 votes
0 answers
126 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
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3 votes
0 answers
181 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
3 votes
0 answers
109 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
3 votes
0 answers
505 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
3 votes
0 answers
126 views

Some Questions regarding the fine Shafarevich-Tate group

The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes ...
debanjana's user avatar
  • 1,161
3 votes
0 answers
104 views

Interpretation or application of this analog of minimal polynomial

Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I ...
PrimeRibeyeDeal's user avatar
3 votes
0 answers
302 views

bounds for the covering radius (or diameter of the Voronoi cell) of a lattice coming from a number field

Let $K$ be a number field and $O$ an order in $K$. Denote the real embeddings of $K$ by $\sigma_1,\dots,\sigma_s$ and the non-real complex embeddings of $K$ by $\sigma_{s+1},\overline{\sigma_{s+1}},\...
Daniel Krenn's user avatar
2 votes
0 answers
116 views

Imaginary quadratic fields with prime class number

Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$. In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write, "Since $h_K = p$, there ...
matt stokes's user avatar
2 votes
0 answers
117 views

$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ 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 ...
Duality's user avatar
  • 1,406
2 votes
0 answers
103 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
2 votes
0 answers
137 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
2 votes
0 answers
99 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
147 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
2 votes
0 answers
122 views

How do elliptic units generate the module of Euler systems over abelian extensions of imaginary quadratic fields?

I am trying to undesrtand the analogy between the Euler systems over abelian extensions of the rationals and the Euler systems over abelian extensions of imaginary quadratic fields. As Soogil Seo ...
Ash's user avatar
  • 99
2 votes
0 answers
110 views

Iwasawa's results about relation between Galois cohomology and principal factorization

Let $K$ be a Galois number field with Galois group and units group $G$ and $U$, respectively. How we can relate the first cohomology group $H^1(G,U)$ to principal factorization in K? I'd try to find ...
A. Maarefparvar's user avatar
2 votes
0 answers
296 views

Definition of Euler system of cyclotomic units

I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt: Let $F=\mathbb{Q}(\mu_m)$. Let $\mathcal{I}(m)$ = {positive square free integers divisible only by ...
Ash's user avatar
  • 99
2 votes
0 answers
91 views

Elementary Iwasawa module

Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?
dekster's user avatar
  • 21
2 votes
0 answers
88 views

Cubic extensions of number fields and their local nature

Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$. Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^...
user116950's user avatar
2 votes
0 answers
204 views

Is there a result that connects the discriminant of an order to the discriminants of the generators of an integral basis for the order?

Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\...
Samuel Hambleton's user avatar
2 votes
0 answers
47 views

Binary quadratic form which takes values that are rational time a square

Let $\mathbb{K}$ be a totally real number field, $a$ an element of $\mathbb{K}$. Consider the quadratic form $q(X,Y) = X^2-aY^2$ and assume that for every number $c$ represented by $q$, if $c$ has the ...
boumbadaboum's user avatar
2 votes
0 answers
87 views

Minimal diameter of a class in a number field

Let $\mathbb{K}$ be a number field of degree $n$, let $A = \mathcal{O}(\mathbb{K})$ be the ring of integers and consider the Minkowski embedding $\mathbb{K} \rightarrow \mathbb{R}^n$ given by $x \...
Preda's user avatar
  • 375
2 votes
0 answers
550 views

Valuation topology vs modified valuation topology

Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
Chilote's user avatar
  • 586
2 votes
0 answers
160 views

relation between class number of an algebraic number field and its Galois closure

I read some results of C. Parry, F. Lemmermeyer and etc, about the class number formula in the special number fields, for example for any pure number field of a prime degree. Now I want to investigate ...
A. Maarefparvar's user avatar
2 votes
0 answers
342 views

Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that $\mathbf{C}$ is ...
Ofra's user avatar
  • 1,603
2 votes
0 answers
489 views

Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of $Q(...
Angel del Rio's user avatar
2 votes
0 answers
136 views

Lowest degree polynomial with integer coefficients yielding $1/\sqrt{2^n}$

Let $x = \cos(\pi/8) = \frac{1}{2} \sqrt{2+\sqrt{2}}$ and $y = \sin(\pi/8) = \frac{1}{2} \sqrt{2-\sqrt{2}}$. What is the lowest degree polynomial $p(x,y)$ with integer coefficients such that $p(x,y) = ...
StephenJ's user avatar
  • 315
2 votes
0 answers
246 views

Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers. I have a question concerning the torsion of elliptic curves over number fields. Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
Chema Tornero's user avatar
2 votes
0 answers
194 views

Comparing ideal class numbers of different orders

Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X]/(P)$. (In general, $...
Oblomov's user avatar
  • 2,501
1 vote
0 answers
73 views

Inflation-restrction sequence for maximal $S$-ramified extension

Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension. There is a inflation-restriction exact sequence, $0\to H^1(Gak(L/K), ...
Duality's user avatar
  • 1,406
1 vote
0 answers
148 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 ...
M. Karatarakis's user avatar
1 vote
0 answers
38 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
1 vote
0 answers
74 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
  • 515
1 vote
0 answers
80 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
83 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