Questions tagged [class-field-theory]
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64 questions
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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 $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...
- 895
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
- 411
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Reference for: power residue symbols are Hecke characters
Notation.
Let $n$ be a positive integer, let $\mu_n\subseteq \mathbb C$ be the set of $N$-th roots of unity, let $K$ be a number field containing $\mu_n$, let $R$ be the ring of integers of $K$, let $...
- 1,384
4
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Examples of norm forms where the numbers represented can be readily described
In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but ...
- 25.7k
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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 ...
- 54.6k
4
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Polynomial equations in many variables have solutions (Lang 1952 paper)
I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...
- 87
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2
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The kernel of the global class field theory homomorphism
Let $K$ be a finite extension of $\mathbb{Q}$. Then there is a surjective homomorphism $\theta:C_K\to G_K^{ab}$ from the idele class group to the abelianization of the absolute Galois group of $K$ (...
user145520
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Unramified extensions of a given degree
Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?
EDIT: If not then under what conditions on $K$,...
- 1,209
2
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1
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333
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CM Elliptic Curves and a result concerning ray class fields
Let $K$ be an imaginary quadratic field and suppose that $E/K$ has complex multiplication by $\mathcal{O}_K$. Let $\psi$ be the Hecke character associated with $E$ and $\mathfrak{f}$ its conductor (i....
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Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
- 14.2k
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Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?
In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
- 6,180
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2
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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 $\...
- 5,966
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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 ...
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Which properties determine the uniqueness of the local Artin map?
Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
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