# Questions tagged [class-field-theory]

The class-field-theory tag has no usage guidance.

287
questions

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### Simplification of links between idele class group and étale cohomology

I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...

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97 views

### Central division algebras over $ \mathbb{Q} $

Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...

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126 views

### Construction of genus class fields

Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...

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

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**1**answer

119 views

### Centralizer of the absolute Galois group of a number field

By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...

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132 views

### Are class numbers of number fields with prime degree often $1$?

I have taken a look at the class number statistics of the L-functions and Modular Forms Database:
https://www.lmfdb.org/NumberField/stats, table "Distribution by class number".
It appears ...

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**2**answers

278 views

### Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...

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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 ...

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**1**answer

103 views

### Are the maximal cyclotomic field contained in a number field and its Hilbert class group the same?

Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(...

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437 views

### how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?

I want to construct the Hilbert class field of $K=\Bbb Q(\zeta_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.

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73 views

### Relation between the Selmer group and the ideal class group

Let $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?

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143 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-...

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89 views

### $0$-th Galois cohomology with topological Milnor K-groups coefficients

In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...

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83 views

### Totally ramified extensions of p-adic fields

Let $\mathbb{Q}_p$ denote the field of p-adic numbers. For a prime number $q$ ($\neq p$), does exist a totally ramified extension $K/\mathbb{Q}_p$ with Galois group isomorphis to $\mathbb{Z}_q \times \...

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829 views

### Set of quadratic forms that represents all primes

A SPECIFIC CASE:
Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$.
If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...

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69 views

### Existence of cyclic extensions ramified at only one prime

For an arbitrary odd prime p is there exists a cyclic extension $K/\mathbb{Q}$ of degree p with a prime power discriminant?

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129 views

### Does K and its Hilbert class field have same conductor?

Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K.$
Definition: (conductor of a abelian number field) Let $K$ be a number field with the abelian Galois group over $\Bbb{Q}....

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213 views

### Etale cohomology and Kummer theory

If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...

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117 views

### Field of algebraic functions

We assume $K$ as a field of characteristic zero. By a field of algebraic functions of one variable over $K$ we mean a field $R$ satisfying $R=K(x,y)$ with $x$ being transcendental over $K$, and $R$ is ...

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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 ...

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**1**answer

135 views

### What are conditions to satisfied by rational prime p so that every prime lying above p is a prime of order 1 and generates class group?

I was reading a paper on Euclidean ideals by H Graves and M. Ram Murthy. I have a problem in understanding one of the claims.
setup
Let $K$ be a number field and $H(K)$ is its Hilbert class field. ...

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274 views

### Artin reciprocity via Shimura varieties

The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a ...

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259 views

### Euler Systems and Coleman’s Conjecture

I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...

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87 views

### When is the natural map of Tate cohomology an isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...

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69 views

### Maximal order of $x^n-d$ and its dependence on $d$

It's well known that the structure of the maximal order of $\mathbb{Q}[\sqrt{d}]$ depends on $d$ modulo $4$: (assuming $d$ is squarefree), the maximal order is $\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\...

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194 views

### Topological structure on higher dimensional local fields

Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something ...

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57 views

### Image of extension ideal classes homomorphism in ideal class group under Artin map in class field theory

Let $K/P$ be a finite extension of number fields and $\epsilon_{K/P}:[\mathfrak{a}] \in Cl(P) \rightarrow [\mathfrak{a}.\mathcal{O}_K]\in Cl(K)$ be the ideal class transfer homomorphism. It's well ...

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484 views

### Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...

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1k views

### Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...

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119 views

### Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of
non-Archimedean local fields where I can find proofs of the following claims about
finite extensions $L/K$ of non-Archimedean local ...

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221 views

### How is class of composition of two quadratic fields is related class numbers of quadratic field?

Let $K_1=\Bbb Q(\sqrt{d_1})$ , $K_2=\Bbb Q(\sqrt{d_2})$ and $K=\Bbb Q(\sqrt{d_1},\sqrt{d_2})$.Suppose $h_1,h_2,h$ be class number of $K_1,K_2,K$ respectively.
(i) Can we express $h$ in terms of $...

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103 views

### Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?

In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...

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103 views

### The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...

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204 views

### Computing the class group of a quadratic function field

I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...

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159 views

### Hilbert class field tower

Let $K$ is a number field,and $H_{K}^{i},i=1,2,\cdots$ be its Hilbert class field tower,suppose it is finite,and let $L=H_{K}^{n}$ is the top of the tower. Must $L$ be galois over $K$?

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192 views

### Henselian valued fields for characteristic $0$: a characterization

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property:
$...

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**1**answer

133 views

### class group size of cyclotomic field subextension

In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension.
What is the best known upper bound for the size of its class group, $\text{...

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64 views

### Anticyclotomic extensions via ideles

Let $ K $ be an imaginary quadratic field with ring of integers $ \mathcal{O} $. Let $ \mathcal{O}_{n} = \mathbb{Z} + n \mathcal{O} $ be the order of conductor $ n $. There is an associated extension $...

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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 $K$ (...

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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 ...

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### $p$-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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### On the determination of ambiguous ideal class of the extension $\mathbb{Q}(\zeta_5,\sqrt[5]{m})/\mathbb{Q}(\zeta_5))$

let $L=\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ and $K=\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic fields, we now that $[L:K] = 5$ and $ GAl(L/K) =\langle\sigma\rangle$ so we call $\mathcal{A}$ an ambigous ...

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### The field generated by the torsion points of an elliptic curve

Let $E$ be an elliptic curve with complex multiplication by an order $\mathcal O$ in an imaginary quadratic field $K$. Let $H=K(j(E))$ and $$L_N=K(j(E),E[N])=H(E[N]).$$
It is not hard to prove that
$...

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59 views

### The completion of a ray class field

I'm reading some papers doing computations on global class field theory.
And the class field theory in those papers is ideal-theoretic.
Here is a question.
Given a base field $k$ and a modulus(cycle)...

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474 views

### How many ways are there to teach class field theory?

I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...

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134 views

### Using the Hilbert symbol to find nice field extensions

Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...

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44 views

### generator of ring class field extension

everyone! I have another questions. Let $K=\mathbb{Q}(\sqrt{-3})$ be an imaginary quadratic field and let $p\equiv 8\mod 9$ be a prime. Denote $H_{3p}$ and $H_{p}$ for the ring class field of $K$ with ...

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108 views

### Values of Grössencharacter attached to CM elliptic curve

I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-...

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### Definition of a Lubin Tate group

Let $L$ be a $p$-adic number field, $\mathcal{O}_L$ its ring of integers, $\pi$ a uniformizer of $L$ and $q$ the cardinality of its residue field.
Let $\varphi(t)\in \mathcal{O}_L[[t]]$ be a ...

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170 views

### Abelianess of $K(j(E))$

Let $E$ be an elliptic curve with CM by an order in the imaginary quadratic field $K$. Is there some easy way how to prove that the extension $K(j(E))/\mathbb Q$ is abelian?
Update
In general, the ...