# Questions tagged [class-field-theory]

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

359
questions

2
votes

1
answer

118
views

### Generators of the ideal class group

Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...

4
votes

1
answer

238
views

### Conductor and local Kronecker–Weber theorem

Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...

2
votes

0
answers

66
views

### $n$-th root of character on local field

Let $F$ be a non-Archidean local field of characteristic 0, and $\zeta_n$ the set of $n$-th roots of unity in the algebraic closure of $F$. Assume $\zeta_n\subseteq F$. Let $\chi:F^\times\to\mathbb{C}^...

4
votes

1
answer

220
views

### How do "Kummer closures" of fields look?

Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...

4
votes

0
answers

112
views

### Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields

Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...

4
votes

0
answers

117
views

### A normal extension of a number field of given degree that does not split over a given set of finite places

Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...

7
votes

0
answers

134
views

### Non-abelian ray class fields for local fields

Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...

2
votes

2
answers

387
views

### Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)

I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...

2
votes

0
answers

94
views

### For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$

Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...

6
votes

0
answers

372
views

### Extensions of p-adic number fields

Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...

2
votes

0
answers

149
views

### Maximal p-extension and pro-p extension

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...

3
votes

0
answers

206
views

### Global class field theory and closure of unit groups

I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...

3
votes

1
answer

296
views

### Where am I going wrong in this interpretation of 1-dimensional geometric class field theory?

I posted this on MSE, but didn't get any responses, so I'm reposting here. I tried to write down an example of the main theorem of geometric class field theory, but I must be misunderstanding ...

5
votes

2
answers

495
views

### Compare with Weber and Hilbert class field

Heinrich Martin Weber and David Hilbert created their own class fields in 1891 and 1897 respectively.
In the past, Weber continued to name $K={Q}(\sqrt{-m}, j(\omega))$, the Kronecker class field of $...

1
vote

0
answers

49
views

### Are integration over restricted direct products only useful for specific functions?

So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\...

2
votes

0
answers

268
views

### Why do we consider characters to $\mathbb{C}$ and not $\mathfrak{p}$-adic or $\mathbb{R}$?

Context: I've been reading Tate's thesis, and in it, we defined the character group for $k^{*}$ and $k^{+}$ for a local field $k$. Here we take the range of the characters to be $S_{1}$ for $k^{+}$ ...

10
votes

2
answers

1k
views

### Why are we defining character groups differently for additive and multiplicative group in Tate's thesis?

Context: I've just started reading Tate's thesis. In it, we start with a local field $k.$ The aim of the section is to describe the structure of the character groups of $k^+$ (the additive group) and $...

2
votes

1
answer

253
views

### Number of imaginary quadratic field with its ideal class group has $\Bbb{Z}/2\Bbb{Z}$ as 2 part

Let $K=\Bbb{Q}(\sqrt{D})$($D$ is a square free negative integer) be a quadratic number field.
Class number (order of ideal class group $Cl_K$ of $K$) is $1$ if only if $D=-2,-3,-7,-11,-19,-43,-67,-163$...

4
votes

1
answer

287
views

### Fields in which $ -1 $ can't be written as sum of two square elements

We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...

4
votes

0
answers

153
views

### The order of the global Galois group

For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...

2
votes

0
answers

87
views

### Compositum of field extensions in context of $\mathbb Z_p$ extension

I had asked this question on stackexchange and I think it is better suited for this site.
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the ...

1
vote

0
answers

85
views

### Examples of $\mathbb{Z}_p$-extensions and two $\mathbb{Z}_p$-extensions with a "nontrivial" intersection

Let $k$ be a number field (a finite extension of $\mathbb{Q}$). Let $p$ be a prime. By saying "$\mathbb{Z}_p$-extensions", we mean Galois extensions $K/k$ of Galois group isomorphic to $\...

3
votes

0
answers

121
views

### Cohomology of local fields in positive characteristic

It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...

2
votes

0
answers

108
views

### Can global fields be defined as certain topological fields like local fields?

It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...

4
votes

0
answers

138
views

### Epstein zeta function for non-fundamental discriminant to L-series

Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...

1
vote

0
answers

200
views

### What is a definition of $A(P_v)$ in the definition of Brauer-Manin obstruction?

This is a question related to the definition of Brauer-Manin obstruction.
Let $K$ be a number field. $X/K$ be an algebraic variety over $K$.
Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...

3
votes

1
answer

324
views

### Local Tate duality for F-vector space

A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...

4
votes

0
answers

161
views

### Does $p$-adic Baker theorem holds in the given case?

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $U_p$ be the units $(1+\mathfrak{m})$ of $\...

5
votes

1
answer

272
views

### Surjectivity of the norm of units in Galois extensions ramified exactly at one finite prime

Let $L/K$ be a finite Galois extension of number fields that is ramified exactly at one finite prime and is unramified at all infinite primes. Let $U_K$ and $U_L$ denote the units of the ring of ...

0
votes

0
answers

197
views

### What is the conductor of $K(\sqrt{2})$ over $K$?

Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...

17
votes

4
answers

2k
views

### What's the use of group cohomology for class field theory?

I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...

4
votes

1
answer

293
views

### Class numbers of cyclotomic fields and their maximal totally real subfields

Let $\zeta_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. I am trying to prove ...

3
votes

0
answers

148
views

### relating class number and narrow class number of a real field

I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...

2
votes

1
answer

112
views

### Ramification of primes and order of $\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of ...

8
votes

2
answers

392
views

### Image of the norm map for Artin-Schreier extensions

Let $k$ be a local field of characteristic $p$ and $\omega \in k$ a uniformiser. Consider the Artin-Schreier extension $L_n = k[x]/(x^p - x - \omega^n)$ for each $n \in \mathbb{Z}$.
Is there an ...

2
votes

0
answers

110
views

### Lubin--Tate formal group construction in local class field theory using group cohomology

Let $K$ be a non-archimedean local field of characteristic 0. Fix a uniformiser $\pi$ and an algebraic closure $\bar{K}$. The theory of Lubin--Tate formal groups gives an explicit construction of the ...

1
vote

0
answers

154
views

### Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$

$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...

1
vote

0
answers

98
views

### CM-fields and ideal theoretic ray class group

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers
and $S$ a finite subset of the real places. Let $\mathfrak{m}
\subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...

1
vote

2
answers

147
views

### The $ 1 \operatorname{ mod } \mathfrak{m}$ congruence relation in ray $ P^{\mathfrak{m}}$ of the ideal theoretic ray class group

Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a
subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...

2
votes

1
answer

231
views

### A question about unramified quadratic extension of number field

Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?

3
votes

0
answers

74
views

### Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients

I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...

3
votes

1
answer

165
views

### Knot group of a field extension

Notation:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}_L^\times$, ideles of $L$
$N_{L/K}$, the norm map
The knot group of an ...

4
votes

1
answer

329
views

### What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...

0
votes

1
answer

390
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 ...

2
votes

0
answers

98
views

### Artin reciprocity for function fields vs number fields

I have a naive question about the difference of Artin reciprocity in the number field versus function field case:
In the number field case, the double quotient we look at is $$K^\times \backslash \...

1
vote

0
answers

100
views

### Rayclass group and Hilbert class group, $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/\mathfrak{a})^\times$

Let $K$ be an imaginary quadratic field with class number $1$. Let $\mathfrak{a}$ be an ideal of $ \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.
Why ...

-1
votes

1
answer

170
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 ...

4
votes

1
answer

150
views

### Existence of lift of (local) Artin map

In a comment to this question, David Loeffler asked if one can show that the (local) Artin map
$$K^\times \to G_K^{ab}$$
does not have a lift to $G_K$. Probably this wouldn't be canonical, but I can't ...

2
votes

0
answers

90
views

### Narrow class number of a the maximal totally real number field inside a cyclotomic field

I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd.
By ...

5
votes

0
answers

197
views

### Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory

I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...