Questions tagged [class-field-theory]

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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 ...
Rashad Ek's user avatar
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 ...
Yijun Yuan's user avatar
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}^...
Windi's user avatar
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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 ...
Theo Johnson-Freyd's user avatar
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 ...
Sebastian Monnet's user avatar
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\...
Mikhail Borovoi's user avatar
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) ...
Riccardo Pengo's user avatar
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 ...
Hilbert Jr.'s user avatar
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 ...
Sebastian Monnet's user avatar
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 ...
Eric's user avatar
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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 ...
Rellw's user avatar
  • 21
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 ...
Tim's user avatar
  • 85
3 votes
1 answer
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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 ...
oggledog's user avatar
  • 133
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 $...
pokssin's user avatar
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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_{\...
Rits's user avatar
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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 $k^{+}$ ...
Rits's user avatar
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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 $...
Rits's user avatar
  • 133
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 $D=-2,-3,-7,-11,-19,-43,-67,-163$...
Duality's user avatar
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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 ...
Sky's user avatar
  • 913
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 ...
Nobody's user avatar
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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 ...
mathemather's user avatar
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 $\...
Hetong Xu's user avatar
  • 547
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)...
aspear's user avatar
  • 31
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 ...
Z Wu's user avatar
  • 340
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 $$\...
pisco's user avatar
  • 331
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)=\...
Duality's user avatar
  • 1,357
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 ...
user14411's user avatar
  • 183
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 $\...
MAS's user avatar
  • 852
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 ...
Ehsan Shahoseini's user avatar
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 $\...
SUNIL PASUPULATI's user avatar
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 ...
Daniel Miller's user avatar
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 ...
did's user avatar
  • 595
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 $...
did's user avatar
  • 595
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 ...
Ehsan Shahoseini's user avatar
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 ...
Daniel Loughran's user avatar
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 ...
User0829's user avatar
  • 1,378
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 ...
user267839's user avatar
  • 5,766
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 $\...
user267839's user avatar
  • 5,766
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 $\...
user267839's user avatar
  • 5,766
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 $K$?
ayoub-chess's user avatar
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 ...
JeeheBo5's user avatar
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 ...
Tristan Phillips's user avatar
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 ...
David Schwein's user avatar
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 ...
user492144's user avatar
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 \...
curious math guy's user avatar
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 ...
Duality's user avatar
  • 1,357
-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 ...
user491084's user avatar
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 ...
curious math guy's user avatar
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 ...
did's user avatar
  • 595
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 ...
youknowwho's user avatar

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