Questions tagged [class-field-theory]

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global and local fields.

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Another question on 1-cocycles (explicite computation of 1-cocycles)

I want to calculate 1-cocycles, as they are described in Neukirch's Class field theory. In what follows the Galois group of $L/\mathbb{Q}$ is acting on $L^{\times}$. The first comment in this question,...
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A very simple problem on 1-cocycles

Edit: According to the comment of Mindlack this special question is solved, but yet I have no idea if my calculations lead to another number $r\neq 0$. And in the case $r\neq 0$, I don't know how ...
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Explicit Local Fundamental Class

Let $L/K$ be a Galois extension of local fields of degree $n:=[L:K]<\infty$ with Galois group $G:=\operatorname{Gal}(L/K)$. In short, my question is as follows. Is there an explicit computation of ...
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Compatibility and consistency between two definitions of cohomology in two books (about coboundary operators and 1-cocycles and computing cohomology)

I was reading cohomology from Neukirch's book, and there he referenced to Hall's book. The two approaches are almost the same (are they not?), and they should give us the same results (cohomology ...
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Proposition 10.4 in Neukirch's Algebraic Number Theory

I am stuck on a small detail in the proof of Proposition 10.4(iii) from Chapter VII of Neukirch's Algebraic Number Theory. For a Galois extension of number fields $L|K$ and a representation $(\rho, V)$...
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Maximal abelian $p$-extension unramified away from $p$

I am having trouble seeing how class field theory implies the following isomorphism. Let $F_n=\mathbb{Q}(\mu_{p^{n+1}})$, let $L_p/F_n$ be the maximal unramified abelian $p$-extension (i.e. the $p$-...
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What is the Weber class polynomial?

I came across the concept of the Weber class polynomial, and it is referenced back to Cox's book, "Primes of the form $x^2+ny^2$". But as far as I searched there I didn't find anything there,...
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What is the Galois group of narrow Hilbert class field over Hilbert class field?

I am trying to solve the following question: Let $K$ be a number field, $H'$ and $H$ be the narrow Hilbert class field and Hilbert class field respectively. Let $O^*_{K,+}\subset O^*_{K}$ be the group ...
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Subgroups of Weil-Deligne group

Let $K$ be a $p$-adic field, i.e. a finite extension of $\mathbb{Q}_p$. As I understand it, when studying Galois representations of $K$, in particualr $p$-adic ones, the "correct" thing to ...
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For each number field $K$ with place $\mathfrak{p}$, prove that $K_{\mathfrak{p}}^{\mathrm{ab}} = K_{\mathfrak{p}} K^{\mathrm{ab}}$

As described in the title, I want to show that for each number field $K$ with place $\mathfrak{p}$, $K_{\mathfrak{p}}^{\mathrm{ab}} = K_{\mathfrak{p}} K^{\mathrm{ab}}$. Here $K_{\mathfrak{p}}$ is the ...
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Let $K$ be a local field and $K'$ be it's finite extension. Then, why ${K'}^{nr}/K^{nr}$ is finite extension?

Let $K$ be a local field and $K'$ be it's finite extension. Then, why ${K'}^{nr}/K^{nr}$ is finite extension ? Unramified extension of local field corresponds to extension of residue field. So, I can ...
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Inertia group $I_K$ is isomorphic to $Gal(\bar K /K^{nr})$ as a group

Let $K$ be a local field. Then, elements of $Gal(\bar K /K)$ which induces identity map on residue field forms group, and the group is called inertia group of $Gal(\bar K /K)$, and written like $I_K$....
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In what kind of situation, ring of integers of $\mathbb{Q}_p(α)$ is $\mathbb{Z}_p[α]$?

In what kind of situation, ring of integers of $\mathbb{Q}_p(α)$ is $\mathbb{Z}_p[α]$? For example, if $α$ is primitive root of unity, then it holds. But it does not hold in general, for example, ...
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Definition of inertia group $I_v$

Inertia group of $Gal(\bar K/K)$ is defined as set of elements of $Gal(\bar K/K)$ that act trivially on the residue field $\bar k$. We often denote inertia group of $Gal(\bar K/K)$ by $I_v$. But ...
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Question about p.165 in Local Fields by Cassels

I am trying to understand the assertions made at the end of the paragraph below. I am not sure if I am missing some certain facts from previous chapters, but I do not see the claims made in bold ...
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Solution of $p＝x^2＋y^2$ in $\Bbb Z$ and $\Bbb Q$ and class field theory

$p＝x^2＋y^2$ has solution in $\Bbb Z$ if only if $p＝2$ or $p\equiv 1 \pmod 4$(Using easy calculation of quotient ring), and has solution in $\Bbb Q$ if only if $p＝2$ or $p\equiv 1 \pmod 4$(Using ...
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Finite galois extension over $\Bbb Q_p$ is always abelian?

Is finite galois extension over $\Bbb Q_p$ always abelian ? I often counts number of give degree extension of $\Bbb Q_p$ using local class field theory, but I'm worrying there are counter example of ...
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Extension of $\Bbb Q_p$ is always abelian extension?

Extension of $\Bbb Q_p$ is always abelian extension ? I know the number of $p$-extension degree abelian extension of $\Bbb Q_p$ is $p+1$ from class field theory,but what about $p$-extension degree ...
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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 ...
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Searching for a concept/clue: efficient algorithm to extract columns in a large table/matrix with minimal collisions: a use case of class field theory

Although the algorithmic problem is very generic in nature and there are many (possibly more down to earth) examples of its application, I do not want to suppress the real context of my problem. I ...
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Subgroup of direct product containing direct sum

I was reading Milne's notes on class field theory. On page 22, while giving an example to the fact that every subgroup of $K^{\times}$, where K is a local field of characteristic $p\neq 0$, is not ...
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Three basic questions about class field theory over $\mathbb{Q}$, splitting versus merely having a zero, and finite fields

(Questions 1, 2 and 3 have been answered, but not the extension of 2 to all finite fields. This is more or less formal, and easy to check given the other statements here.) I've been reading about ...