# Questions tagged [class-field-theory]

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

331
questions

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### 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

1k
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### 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

139
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### 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 ...

2
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0
answers

109
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### 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

59
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### 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

307
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### 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

98
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### 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

129
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### 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
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0
answers

75
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### 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

124
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### 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

166
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

57
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

118
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

224
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### 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

128
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### 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

83
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### 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 \...

2
votes

0
answers

83
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

137
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### 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

127
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

56
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

130
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### 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 ...

4
votes

0
answers

135
views

### What are the applications of $\lambda$-rings to class field theory?

In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...

1
vote

0
answers

113
views

### Units in residue classes modulo prime ideal

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...

2
votes

0
answers

126
views

### About Lubin Tate extensions

https://math.stackexchange.com/questions/4473761/about-lubin-tate-extensions
I know this is a very low-level question because it is about a probable typo and a definition. But I asked this question on ...

2
votes

1
answer

125
views

### Constructing Ray class fields of $\mathbb{Q}(i)$ and division points of lemniscate

I've heard that it was Abel who first constructed some of (or all of?) Abelian extensions of $\mathbb{Q}(i)$ using division points of lemniscates. Can we construct the Ray class fields of $Q(i)$, or ...

5
votes

1
answer

230
views

### Do there exist elliptic curves over $H_K$ having everywhere good reduction and CM by $\mathcal{O}_K$?

For $K$ a number field, denote by $\mathcal{O}_K$ its ring of integers and by $H_K$ its Hilbert class field.
For which imaginary quadratic field $K$ does there exist an elliptic curve $E$, defined ...

3
votes

1
answer

257
views

### Lang's proof concerning ray class fields of imaginary quadratic number fields

Crosspost from Math.SE as I did not receive an answer there:
In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...

6
votes

0
answers

188
views

### Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$

$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...

6
votes

1
answer

403
views

### Chebotarev density theorem and pure weight local systems

How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper.
Let $U$ be a scheme of finite type over $\mathbb{F}_q$. Let $\...

2
votes

0
answers

121
views

### Artin map and profinite completion of the idèles

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...

0
votes

0
answers

112
views

### Tate cohomology and cup product: functoriality in $G$

Let $G$ be a finite group, and let $A, B$ be $G$-modules.
See Atiyah and Wall [AW] for the definition of the Tate cohomology groups $H^q(G,A)$ for all $q\in\mathbb Z$
and of the cup product pairings
\...

2
votes

0
answers

92
views

### Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by
$$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$
and
$$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$
Question:
Given a ...

0
votes

1
answer

254
views

### Coboundary operators, 1-cocycles and computing cohomology

My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago on MathSE
and I set a bounty on it, but I didn't ...

3
votes

3
answers

426
views

### Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected?

I was reading this question The connected component of the idele class group but I am very confused about the structure of the solenoids $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\...

5
votes

0
answers

300
views

### The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field,
and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$.
By local class field theory, there is a canonical isomorphism
$$...

7
votes

1
answer

235
views

### Explicit cocycles for the first Galois cohomology of a $p$-adic torus

Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...

2
votes

0
answers

119
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### Conditions for being an entry in a trace compatible sequence

$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal ...

8
votes

1
answer

232
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### A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...

12
votes

1
answer

463
views

### Finite Galois module whose Ш¹ is nonzero?

In algebraic number theory, we constantly make use of the nine-term Poitou-Tate sequence: Let $K$ be a number field and $M$ a finite $K$-Galois module. Then we have the nine-term exact sequence
$$
H^0(...

4
votes

1
answer

224
views

### How to calculate genus number of number field using sage?

I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number ...

2
votes

0
answers

89
views

### Is there data base of quadratic fields which have abelian Hilbert class field?

I did not find it in LMFDB. However, I am looking for a Database; please feel free to navigate me to any paper, which might be helpful.
I really appreciate any help you can provide.

4
votes

0
answers

225
views

### Explicit invariant map in local class field theory

Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...

4
votes

1
answer

208
views

### A Kummer exact sequence involving $\mu_\infty$

Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...

2
votes

1
answer

131
views

### Field extension corresponding to a quotient of units of local fields

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $\mathcal{O}$ be its ring of integers and $\mathfrak{m}$ the maximal ideal. Pick a uniformiser $\pi$. The construction using theory of Lubin--Tate ...

4
votes

0
answers

208
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### Reference request for Kummer-Artin-Schreier-Witt theory

I cannot find the following 4 papers by Sekiguchi–Suwa in their works on Kummer–Artin–Schreier–Witt theory:
On the unified Kummer–Artin–Schreier–Witt theory, Prépublications du laboratoire de ...

3
votes

0
answers

174
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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 ...

3
votes

0
answers

170
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 ...

2
votes

0
answers

141
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 ...

4
votes

1
answer

175
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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 $N_{H/K}: I_H \to P_K$ ...

3
votes

1
answer

156
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### 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-...