# Questions tagged [number-fields]

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191
questions

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70 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 \...

**3**

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68 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 $N_{H/K}: I_H \to P_K$ ...

**2**

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74 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 $P$) ...

**2**

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101 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 ...

**10**

votes

**1**answer

443 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 ...

**1**

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94 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?

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79 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 ...

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71 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(...

**4**

votes

**2**answers

158 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 $k \in \{ \pm 1, \pm 2, \pm 4 \}$, ...

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55 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 $...

**8**

votes

**1**answer

213 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 $\exists\forall$-...

**5**

votes

**1**answer

121 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 $\...

**3**

votes

**2**answers

222 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 ...

**2**

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77 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 ...

**2**

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170 views

### Looking for a paper of Lagarias and Odlyzko

I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while:
Effective versions of the Chebotarev density theorem, ...

**8**

votes

**1**answer

223 views

### Standard conjecture on u-invariants?

This is well beyond my expertise, but I just learned some of the history behind
$u$-invariants of fields $F$,
where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution,
but $u(F)...

**0**

votes

**1**answer

142 views

### English translation of Hasse's “Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage”

I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...

**2**

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78 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 ...

**1**

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103 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 ...

**0**

votes

**1**answer

120 views

### Logarithmic Weil height

Let $a_0,\cdots,a_n$ be algebraic integers. Is $h(a_0,\cdots,a_n)\le\max_{0\le i\le n}\log(\max(1,|a_i|))$ where $h(a_0,\cdots,a_n)$ denotes the logarithmic Weil height?
Thanks in advance.

**1**

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151 views

### Do roots of polynomial with coefficients in a CM field lie in a CM field?

This is something that I have been thinking about for a while now, not sure if it is standard (or even true at all) or not:
Let $K/ \mathbb Q$ be a CM number field, that is, it is closed under complex ...

**9**

votes

**1**answer

262 views

### Galois embedding question for dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Then all the quotients of $D_n$ are dihedral as well, and of the form $D_k$ with $k \mid n$. So for a field $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\...

**1**

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**1**answer

173 views

### $G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$

I know that if $K/\mathbb Q$ is a finite Galois extension (i.e. a Galois number field), then for any prime $(p)\subseteq \mathbb Z$, the Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ acts ...

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41 views

### prime decomposition in even dihedral extensions

Let $L/K$ be a finite extension of number fields of degree $n$ with $n$ an even integer such that the normal closure of $L$ has the Galois group isomorphic to $D_n$, the dihedral group of order $2n$. ...

**3**

votes

**1**answer

147 views

### Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number

Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true?
For every $\alpha \in K^\times$ there exists $\...

**2**

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75 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?

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168 views

### Computing the kernel of some Artin-Map

let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(i)$. $\mathcal{O}_K$ is the ring of integers of K.
I would like to show that the kernel of the Artin-Map $\phi_{L/K}: I_K \rightarrow Gal(L/K)$ is $P_K$, ...

**-1**

votes

**1**answer

58 views

### Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]

I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...

**2**

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69 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^...

**3**

votes

**1**answer

185 views

### A question on p-rationality of number fields

Let $p$ be an odd regular prime and $F$ be a $p$-rational number field containing $\mu_p$. Equivalently, there is a unique prime $\mathfrak{p}$ above $p$ in $F$ and the $p$-class group is generated by ...

**3**

votes

**1**answer

296 views

### Integrality of Atkin-Lehner operator for $\Gamma_1(N)$

A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...

**1**

vote

**1**answer

425 views

### Sets of primes with a given Frobenius conjugacy class

Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...

**8**

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**1**answer

256 views

### Commutator subgroup of the absolute Galois group - a closed subgroup

Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...

**3**

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**2**answers

300 views

### The kernel of the global class field theory homomorphism

Let $K$ be a finite extension of $\mathbb{Q}$. Then there is a surjective homomorphism $\theta:C_K\to G_K^{ab}$ from the idele class group to the abelianization of the absolute Galois group of $K$ (...

**1**

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**1**answer

352 views

### A type of principal ideal theorem of class field theory for ramified primes

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Also let $p$ be a prime number, $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ and $\zeta_{m}$ be a primitive mth root of ...

**1**

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98 views

### Octonion algebras over number fields [closed]

Is there any textbook or paper about Arithmetic of Octonion Algebras or Octonion Algebras constructed over number fields? I know J. Voight book and K. Martin notes about quaternion algebras but I was ...

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121 views

### $p$-torsion of class groups

Let $p$ be a fixed odd prime and $\ell$ be another prime such that $\ell \equiv 1 \pmod{p}$.
Consider the number field $\mathbb{Q}(\zeta_p)$ and its extension $\mathbb{Q}(\zeta_p, \zeta_\ell)$. Note ...

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148 views

### Matrix factorizations over $GL_2$ of a real quadratic ring of integers

tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...

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78 views

### pari/gp “bnfisintnorm” as poor man (quadratic) Thue equations solver?

For simplicity explaining only the quadratic case.
Given integers $n,m$, pari/gp "bnfisintnorm" finds $X,Y$
such that $X^2+n Y^2=m$ working in the number field
with defining polynomial $x^2+...

**1**

vote

**1**answer

153 views

### How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?

More specifically, I need the degree $d$ to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in $Z_2$, i.e., $x^{1024}+1$, which is equal ...

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120 views

### Terminology about ramification

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $...

**2**

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**3**answers

242 views

### Mahler measures of values of polynomials

Let $K\ne \mathbb{Q}$ be a number field, let $\alpha\in \mathcal{O}_K$ and let $f(X)\in \mathcal{O}_K[X]$. Denote the Mahler measure by $M$.
Is there any known result about the comparison of the ...

**3**

votes

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159 views

### Dihedral extension of $\mathbb Q$ with small discriminant

Let $K$ be a fixed quadratic number field, say $K=\mathbb Q(\sqrt 5)$. For any integer $n \geq 3,$ I would like to build a number field $D_n$ such that $D_n/\mathbb Q$ is Galois, with Galois group ...

**12**

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**1**answer

1k views

### Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:
Lemma: if L/K is an abelian ...

**3**

votes

**1**answer

229 views

### Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...

**9**

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196 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 ...

**2**

votes

**1**answer

115 views

### why the division field of an abelian variety contains a cyclotomic field?

Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group
of points of $A$ that are annihilated by $\ell$, the ...

**11**

votes

**1**answer

457 views

### Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...

**3**

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100 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 ...

**2**

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116 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 $\...