Questions tagged [class-field-theory]
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382 questions
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Lubin-Tate vs cohomological approach to local CFT
Local class field theory ("local CFT") can be developed in various ways, among them is a cohomological approach and an explicit approach due to Lubin and Tate (both can be found in Milne's CFT notes ...
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class numbers of $\mathbf{Q}(2^{1/n})$
Calculating the class numbers of $\mathbf{Q}(2^{1/n})$ for small $n$ always yields $1$. Is it true for an infinite number of $n$s? Does applying Iwasawa theory to the false Tate curve tower $\mathbf{...
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Embedding number fields in fields with class number 1
(Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory ...
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Parity of class number of pure cubic fields
A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure ...
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Image of norm map for local field
Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
What is the image of the norm map $N_{E/F}$?
In particular - ...
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3
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remark in milne's class field theory notes
In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between $Gal(L/K)$ and $C_m/H$ where $H$ is ...
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What is the "reason" for modularity results?
The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? .
I ...
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2
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How do Brauer groups relate to zeta functions?
There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
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2
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1
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Lower bound on the class group of the p-Hilbert class field of an imaginary quadr. field
Let K be an imaginary quadratic field, A(K) its p-class group, and H(K) its p-Hilbert class field. If rk(A(K))=2, a result due to Arrigoni tells us that p^3 divides the order of the class group of H(K)...
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Class number parity in pure cubic number fields
Consider the family of pure cubic number fields
$K = {\mathbb Q}(\sqrt[3]{m})$ for $m = a^3 \pm 3$.
Proposition. If $4 \mid a$ and $m$ is cubefree, then the
class number of $K$ is even.
Proof. Let $...
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answer
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What are the roots of unity in abelian extensions of imaginary quadratic fields?
What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know:
Is $K(\zeta_n)/K$ an abelian ...
- 32.6k
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2
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Why is Class Field Theory the same as Langlands for GL_1?
I've heard many people say that class field theory is the same as the Langlands conjectures for GL_1 (and more specifically, that local Langlands for GL_1 is the same as local class field theory). ...
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Help wanted with Chebotarev condition in characteristic 2
Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
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Where does the principal ideal theorem (from CFT) go?
My impression is that one of the celebrated results of class field theory the principal ideal theorem namely that given a number field $K$ and its maximum unramified abelian extension L, every ideal ...
- 32.6k
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Decomposing anticyclotomic characters
Suppose $K/\mathbf{Q}$ is an imaginary quadratic field and $\chi$ is a finite-order character of $G_K=\mathrm{Gal}(\overline{K}/K)$ which is anticyclotomic, i.e. $\chi^{\sigma}:=\chi(\sigma g \sigma^{-...
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transfer kernels and the Schur multiplier
Let $\Gamma$ be a finite $2$-group, and let $G$ be any subgroup
of index $2$. Moreover, let Ver$: \Gamma/\Gamma' \to G/G'$
denote the group theoretical transfer, and let $M(\Gamma)$ be
the Schur ...
- 32.6k
9
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1
answer
591
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On the structure of the maximal abelian Galois group of a number field
Let $K$ be a number field. I am wondering if the following exact sequence $$1 \longrightarrow[\widehat{\mathcal O}_K^\times] \longrightarrow Gal(K^{ab}/K) \overset{\pi}{\longrightarrow} Cl_K \...
- 32.6k
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1
answer
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Are class numbers encoded in the absolute Galois group of ${\mathbb Q}$?
The absolute Galois group $G_{\mathbb Q}=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$, as a profinite group, encodes a lot of things: the whole lattice of number fields (closed subgroups of finite index), ...
- 13.1k
4
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1
answer
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group theoretical transfer map and its consequences
I'm trying to understand whether there is a sophisticated reason that forces the transfer map to play its role in class field theory or not. Because, at least in Neukirch's proof (at his book ANT) on ...
- 32.6k
12
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answers
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On the relation of special values of motivic L functions and partial zetas
Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this ...
- 2,029
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Locally-trivial-cycles analogy for Arakelov classes instead of ideal classes?
The well known isomorphism:
$$Cl(K) \cong Ker\\ \lgroup\\ H^1(G_K, U) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},U_p) \rgroup$$
is great. ("Visibility of Ideal Classes", Schoof and ...
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Most "natural" proof of the existence of Hilbert class fields
Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...
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7
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2
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732
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Explicit map for Scholz reflection principle
The question is about the specific case of reflection theorems (copied straight from Franz Lemmermeyer's "Class Groups of Dihedral Extensions"):
Let $k^+ = \mathbb{Q}(\sqrt{m})$ with $m\in \mathbb{...
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votes
3
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Kummer generator for the Ribet extension
Let $p$ be an odd prime and let $k\in[2,p-3]$ be an even integer such that $p$ divides (the numerator of) the Bernoulli number $B_k$ (the coefficient of $T^k/k!$ in the $T$-expansion of $T/(e^T-1)$). ...
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votes
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answers
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Analogue of a ring extension splitting in the Kummer case
Background (the Kummer extension case)
Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=...
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2
answers
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Elementary Aspects of Galois Deformation
Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) ...
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2
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Properties shared by number fields with the same normal closure?
While studying some class field theory there was a lot of talk on galois extensions. Of course. When talking about non-galois number fields, usually the text will quickly take the galois closure. At ...
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2
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Algorithm for the class field tower problem?
This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:
Let $K$ be a number field. The (Hilbert) class field tower of $K$ is the ...
- 32.6k
11
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1
answer
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Is there an analog of class field theory over an arbitrary infinite field of algebraic numbers?
Recently, I found a paper by Schilling http://www.jstor.org/pss/2371426, which mentions that for certain infinite field of algebraic numbers there is an analog of class field theory. By infinite field ...
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3
votes
1
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ray class field of rational function field
Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module....
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What's the Hilbert class field of an elliptic curve?
My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some ...
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3
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Topological Langlands?
In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
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