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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49 views

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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42 views

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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1answer
70 views

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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34 views

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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1answer
38 views

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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1answer
45 views

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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46 views

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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1answer
111 views

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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1answer
120 views

On the maximal abelian pro-$p$ extension unramified outside $p$ and the Leopoldt's conjecture

Now I'm struggling on the following problem: Problem: Let $K$ be a number field and $p$ be prime number. Let $M$ be the maximal abelian pro-$p$ extension of $K$ unramified outside $p$. Describe $\...
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1answer
48 views

Are $(K_v)^{ab}$ and $(K^{ab})_v$ the same thing?

I saw the following commutative diagram in the book Number Theory 2: Introduction to Class Field Theory by Kato, Saito, Kurogawa. And the authors says that the rightmost vertical map is the ...
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1answer
55 views

$2^n$-th rational reciprocity laws

Let $p,q$ be odd coprime primes. We are familiar with the quadratic reciprocity law: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{(p-1)(q-1)}{4}}$. This is generalized (see for ...
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1answer
139 views

$\mathbb{Q}(\sqrt{p^*})$ is contained in the ring class field of conductor $p$

Let $K$ be an imaginary quadratic field, $p$ a prime of $\mathbb{Q}$ and $H_p$ the ring class field of $K$ of conductor $p$, i.e. the abelian extension of $K$ with Galois group isomorphic to the class ...
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1answer
62 views

Group of Square classes of $\mathbb{Q}[i]$

I am trying to figure out the structure of the group of square classes $F^{\times}/(F^{\times})^{2}$ where $F=\mathbb{Q}[i]$ (Gaussian numbers). I was trying when a gaussian integers is a perfect ...
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1answer
59 views

Isomorphisms of local and global class field theory

Let $k$ be a number field and $K/k$ be a finite Galois extension of degree $d=[K:k]$ with Galois group $\Gamma=\Gamma_{K/k}$. Let $v$ be a place of $k$, and let $k_v$ denote the completion of $k$ at $...
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27 views

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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2answers
61 views

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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1answer
99 views

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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1answer
65 views

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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66 views

Does $\Bbb Q_p^×$ have index $n$ subgroup?

Let $n$ be arbitrary positive integer. Does $\Bbb Q_p^×$ have index $n$ subgroup? If I could prove this, from local class field theory, I can say $\Bbb Q_p$ has arbitrary degree of extension. I know $\...
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19 views

Index $n$ subgroup of local field exists?

Let $K$ be a local field. Let $n$ be arbitrary positive integer. Does $K^×$ have index $n$ subgroup? If I could prove this, from local class field theory, I can say $K$ has arbitrary degree of ...
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1answer
89 views

Inertia group of $\operatorname{Gal}{(\overline{\mathbb{Q}}_p/{\mathbb{Q}}_p)}$

I want to find inertia group of $\text{Gal}{(\overline{\mathbb{Q}}_p/{\mathbb{Q}}_p)}$ . It is well known that the inertia group is isomorphic to $\operatorname{Gal}{(\overline{\mathbb{Q}}_p/{\mathbb{...
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160 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 ...
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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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38 views

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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83 views

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 ...
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Existence of weil group and cohomology of class fields

I currently reading an article by Tate in "Automorphic forms, representations and L-functions". The book consists of lectures given at Oregon state university in 1977. I am stuck at section (...
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Cohomological interpretation of Tate cohomology quotient in local class field theory

Let $E/L/K$ a tower of finite abelian extensions of local fields, $G = \operatorname{Gal}(E/K), H = \operatorname{Gal}(E/L)$. There is of course a natural quotient in Tate cohomology: $$ \phi_\star: \...
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Maximal Unramified Extension $\mathbb{Q}_p^{nr}$ of $\mathbb{Q}_p$

A question about some reasonings in Example 1.3 from these notes on unramified extensions. Example 1.3. Consider the finite unramified extensions of $\mathbb{Q}_p$. By the above theorem, these are in ...
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Overly complicated proof of the surjectivity of the Artin map

Let $K/k$ be an abelian extension of number fields and $\frak c$ a cycle of $k$ divisible by all primes ramifying in $K$. Then we have the following theorem from algebraic number theory (see Lang's ...
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An isomorphism between a quotient of the idèles of a number field and a quotient of the idèle class group

Let $K$ be a number field, $L/K$ finite Galois, $\Bbb I_K$ and $\Bbb I_L$ their respective idèles and $C_K, C_L$ their idèle class groups. Let $N_{L/K} : \Bbb I_L \to \Bbb I_K$ be the norm on the ...
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The classical statement of class field theory

The classical statement of class field theory is that for finite Galois extension $L/Q$ the following are equivalent: a. $L$ is a class field. b. $L/Q$ is abelian. c. $L \subset Q[\zeta _n]$ for some $...
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40 views

Discriminant of non-degenerated Billinearform is square in Ideal Class Group

Let $A$ be a Dedekind domain, $K= Frac(A)$ its field of fractions and $V$ a $n$-dimensional vector space over $K$. A lattice of $V$ (with respect to ring $A$) is a sub-$A$-module $X$ of $V$ that is ...
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Proof of Global Kronecker-Weber, Global Case

This question comes from Ramakrishnan's "Fourier Analysis on Number Fields" and its proof of Kronecker-Weber. Namely, I do not understand the final lines of the proof found on p. 236. Let us ...
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Ramification in cyclotomic extension

I am trying to understand how primes ramify in $\mathbb Q(\zeta _p)/\mathbb Q$ for $p$ prime. From class field theory, how a prime $q$ ramifies depends only on $q \mod p$. I have following particular ...
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Calculating Ray class Numbers in sagemath?

I got the code of this class from here https://ask.sagemath.org/question/9127/computing-the-order-of-an-ideal-in-a-ray-class-group/ , problem is it gives the error NameError: name 'AbelianGroup_class'...
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Deducing class field theorotic phenomenon without Chebotarev density theorem.

Let $K/\mathbb Q$ be a number field, if there exists a integer $N$ such that for $S \subset \mathbb Z / N\mathbb Z$ $$ \text{Spl} (K/\mathbb Q)=\{ p \ | \ p \mod N \in S \} $$ then $K/\mathbb Q$ is ...
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Questions about the chapter "Abstract class field theory" in Neukirch's "Algebraic Number Theory"

I am trying to learn chapter IV, "Abstract class field theory", from Neukirch's book "Algebraic Number Theory". My problems occur in paragraphs 4 and 5 of that chapter. On two ...
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58 views

Simplification of links between idele class group and étale cohomology

For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global field. I started with the link between étale cohomology and Galois cohomology, $H^i(\...
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1answer
47 views

Open subgroup of a valuation ring in Lemma 2.3 in Milne's Class Field Theory

I'm reading Milne's Class Field Theory. On the page 104, it gives the following lemma: In the lemma, $K$ is a local field, and $\pi$ is the uniformizer of $L$. LEMMA $2.3$ Let $L$ be a finite Galois ...
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1answer
108 views

Unramified subextensions of $\mathbf{Q}(\alpha,\sqrt{-23})$

Let $\alpha$ be a root of the polynomial $f=X^3-X-1$. The following exercise should guide me through the standard example of a Hilbert class field. I showed that the class group of $\mathbf{Q}(\sqrt{-...
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2answers
47 views

Prove that the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{m^2 - 1})$ is $m+\sqrt{m^2 - 1}$ if $m^2-1$ is square free

Prove that the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{m^2 - 1})$ is $m+\sqrt{m^2 - 1}$ if $m^2-1$ is square free I tried taking $a + b\sqrt{m^2-1}$ as an arbitrary unit and ...
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1answer
59 views

Class field theory for $\mathbb{Q}_p$

Let $K$ be a local field and fix an algebraic closure $\bar{K}$. Local class field theory says basically that $L\mapsto N_{L/K}(L^\times)$ is a order-reversing bijection between the finite abelian ...
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1answer
82 views

Proving that the map $k^*/N_{K/k}(K^*) \to \operatorname{Br}(K/k)$ is injective.

I'm trying to understand the proof of the following result in number theory: Let $K/k$ be a cyclic extension and let $\sigma$ be a generator of its Galois group. The map $a\mapsto [(\sigma,a)]$ ...
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1answer
32 views

Frob_p under class field theory for cyclotomic extension of $\mathbb{Q}$

Why does $$p^{-1} \in \mathbb{Q}_p^\times \subseteq \mathbb{A}^\times/\mathbb{Q}$$ go to $Frob_p$ for the cyclotomic extension $\mathbb{Q}(\mu_{p^\infty})/\mathbb{Q}$.

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