# Questions tagged [number-fields]

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206
questions

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### 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 $...

1
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0
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56
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### 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 ...

1
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0
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76
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### 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 ...

2
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0
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65
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### 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 ...

3
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0
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116
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### 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I was reading Eichler's "Allgemeine Kongruenzklasseneinteilungen [...]", Crelle 1938 (one of the main historical references for strong approximation theorems), and I cannot understand one of ...

4
votes

1
answer

156
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### 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 ...

4
votes

1
answer

242
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### 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 ...

3
votes

0
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96
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### 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 $...

4
votes

0
answers

91
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### 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 ...

2
votes

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119
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### 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 ...

3
votes

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### 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}^{...

1
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0
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87
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### 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.
...

9
votes

1
answer

544
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### 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 $\mathcal{O}_K$. ...

2
votes

1
answer

132
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### 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 $...

3
votes

0
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275
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### 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 ...

6
votes

1
answer

284
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### 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 \...

4
votes

1
answer

126
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### 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
votes

0
answers

78
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### 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
votes

0
answers

127
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### 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 ...

13
votes

1
answer

775
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### 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
vote

0
answers

100
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### 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?

0
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0
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82
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### 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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0
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82
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### 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

205
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### 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 $k \in \{ \pm 1, \pm 2, \pm 4 \}$, ...

1
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0
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59
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### 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

224
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### 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

141
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### 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

316
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### 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
votes

0
answers

99
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### 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 ...

1
vote

0
answers

330
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### 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, ...

9
votes

1
answer

321
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### 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

155
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### 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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0
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### 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 ...

2
votes

0
answers

162
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### 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

189
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### 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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0
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### 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

280
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### 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
vote

1
answer

232
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### $G_{\mathbb Q}$ 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 ...

0
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0
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### 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
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169
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### 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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0
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78
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### 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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198
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### 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 $P_K$, ...

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votes

1
answer

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### 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 ...

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0
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### 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^...

4
votes

1
answer

329
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### 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

316
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### 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

452
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### 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
votes

1
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316
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### 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
votes

2
answers

404
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### 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
vote

1
answer

429
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### 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 ...