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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.

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global fields and local fields. It also includes a reciprocity homomorphism which acts from the idele class group of a global field to the Galois group of the maximal abelian extension of the global field.

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Hilbert class field of $\mathbb Q(\sqrt{-39})$

Is the Hilbert class field of $K=\mathbb Q\left(\sqrt{-39}\right)$ is equal to $K\left(\sqrt{-39},\frac{\sqrt{1+\sqrt{13}}}{2}\right)$
Math123
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Computing Hilbert Class Field of a number field

I'm trying to compute the Hilbert class field of the extension $\mathbb{Q}(\zeta_{5}, \sqrt{-43})$. I know that it has class number 7. I would like to show that it is contained in $\mathbb{Q}(\zeta_{215})$.
user513566
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Hilbert 2-Class Field definition

what is a Hilbert 2- class field? As a Hilbert Class field of a number field K is the maximal unramified abelian extension, of K,
R Devadiga
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About the definition of local fields

I read in a book that the definition of local fields is: A field $K$ for which with respect to a discrete valuation $v$, the residue field is finite and $K$ is complete with respect to $v$. However, I have a few questions: Is it the field $K$ which…
user584333
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ramification of infinite place of ${Q}$

Can any one explain me "The infinite place $v=\infty$ of ${Q }$ is unramified in ${Q(\sqrt{2})} $ but is ramified in ${Q(\sqrt{i})}$
Math123
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Conductor of a ray class field.

I am not getting the definition of Conductor of a ray class field. I know the following definition Let $K$ be a number field. The theorems of class field theory tell us that given any modulus $\mathfrak{m}$ for $K$, there is a unique Abelian…
Math123
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Hilbert class field of $\mathbb Q(\sqrt{-5})$ and $\mathbb Q(\sqrt{-10})$

This might be a very silly doubt, as I am making an attempt to understand Hilbert Class field, I would like to know this. In Cohen,"Advanced topics in computational number theory" Hilbert class field for $K=\mathbb Q(\sqrt{-5})$ and $K=\mathbb…
Math123
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Exercises for local class field theory

I'm planning to give a mini course about local class field theory with some people. We made a lecture note based on Neukirch's "Class field theory - Bonn lecture". It seems that it will be great if we put some exercises for group cohomology and…
Seewoo Lee
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What is the essence of Class Field Theory for $\mathbb{Q}$

I have read that the goal of Class Field Theory is to characterize all the abelian extensions of a number field $K$ in terms of parameters of $K$. Let $L|K$ be an abelian extension of number fields. I want to know what are the main theorems of…
learning_math
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Evaluating Artin symbol

Consider the field $K=\mathbb Q(\sqrt{2})$. Let $mp=\frac{(2+\sqrt{2})^{p}-1}{1+\sqrt{2}}$. For the field extension, $K(-1+2\sqrt{2})/K$, and $p\equiv 5\bmod{6}$ how can one show $ \frac{(2+\sqrt{2})^{p}-1}{1+\sqrt{2}}\equiv 1\pmod{-1+2\sqrt{2}…
Math123
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Understanding the isomorphism $\widehat{K^*} \cong {\mathcal{O}^*_K} \times \widehat{\mathbb{Z}}$

Let $K$ be a local field and let $\widehat{K^*}$ and $\widehat{\mathbb{Z}}$ denote the profinite completions of $K$ and $\mathbb{Z}$. As the title suggests I'm having difficulties understanding the isomorphism ${\mathcal{O}^*_K} \times…
coconuthead
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Artin reciprocity

The Artin reciprocity says that if $L/\mathbb{Q}$ is a finite abelian extension with defining modulus $m$, then the sequence of groups $$ 1\to I_{L,m}\to (\mathbb{Z}/m\mathbb{Z})^\times\to Gal(L/\mathbb{Q})\to 1 $$ is exact. I do not quite see why…
user584333
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About the Artin symbol

Let $L$ be a finite abelian extension of $\mathbb{Q}$ and let $m$ be a positive integer such that $L\subset\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $m$-th root of unity. Let $a$ be an integer coprime to $m$. Then the Artin symbol…
user584333
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Some property of norm residue symbol

On the last page of this article, Artin and Hasse used some property of the norm residue symbol $$\left(\frac{\lambda}{A}\right)_K = \left(\frac{\lambda}{n(A)}\right)_{k_\zeta}$$ (under Beweis von Satz 3) but I cannot seem to find any reference for…
An Hoa
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Serge Lang ANT Theorem 7 p. 148

Let $\mathfrak c, \mathfrak f$ be admissible cycles for an extension of number fields $K/k$ with $\mathfrak f$ dividing $\mathfrak c$. If you're not familiar with the terminology, a cycle $\mathfrak c$ is admissible if any $1 + \mathfrak…
D_S
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