Questions tagged [class-field-theory]
The class-field-theory tag has no usage guidance.
382 questions
37
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5
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Tips on cohomology for number theory
I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.
Do people just remember all the rules and ...
6
votes
0
answers
496
views
The Tate-Nakayama theorem and inflation
Let $K$ be a nonarchimedean local field,
and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$.
By local class field theory, there is a canonical isomorphism
$$...
- 14.2k
2
votes
0
answers
110
views
Narrow class number of a the maximal totally real number field inside a cyclotomic field
I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd.
By ...
- 637
7
votes
5
answers
3k
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Number of Galois extensions of local fields of fixed degree
Let $K$ be a local field (of characteristic 0) with (finite) residue field of characteristic $l$ and let $p$ be a prime.
Considering the cases, whether the $p$-th roots of unity are in $K$ and ...
- 841
12
votes
2
answers
605
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Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor
It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
- 9,501
25
votes
2
answers
4k
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Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 26.1k
4
votes
1
answer
580
views
Etale cohomology and Kummer theory
If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
- 63.1k
2
votes
0
answers
256
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About Lubin Tate extensions
https://math.stackexchange.com/questions/4473761/about-lubin-tate-extensions
I know this is a very low-level question because it is about a probable typo and a definition. But I asked this question on ...
7
votes
1
answer
343
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Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...
- 14.2k
4
votes
1
answer
267
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Norm groups of number fields
I came across this proposition in an article about genus class fields.
I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
- 577
2
votes
1
answer
197
views
Field extension corresponding to a quotient of units of local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $\mathcal{O}$ be its ring of integers and $\mathfrak{m}$ the maximal ideal. Pick a uniformiser $\pi$. The construction using theory of Lubin--Tate ...
- 1,428
8
votes
1
answer
552
views
Artin reciprocity via Shimura varieties
The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a ...
- 4,428
2
votes
2
answers
469
views
Relation between the Selmer group and the ideal class group
Let $E/K$ be an elliptic curve defined over the number field $K$. Does exist any relation between the $p$-Selmer groups of $E/K$ and the ideal class group $Cl(K)$ of $K$?
- 437
2
votes
0
answers
159
views
Artin map and profinite completion of the idèles
One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...
- 3,446
4
votes
1
answer
322
views
How to calculate genus number of number field using sage?
I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number ...
- 753
36
votes
1
answer
3k
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Artin reciprocity $\implies $ Cubic reciprocity
I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...
- 1,207
21
votes
3
answers
1k
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The Teichmüller's algebraic interpretation of $H^3$ in group cohomology
In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an ...
10
votes
1
answer
707
views
Tables of class numbers of cyclotomic fields
Does anyone have a table of the class numbers ($h_n$) of cyclotomic fields (upto say, n = 250-300 for $\mathbb Q(\mu_n)$)?
I can find tables for the relative class number ($h_n^-$) in various places ...
- 7,746
17
votes
6
answers
3k
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Reference for learning global class field theory using the original analytic proofs?
I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local ...
- 15.4k
6
votes
1
answer
451
views
How is class of composition of two quadratic fields is related class numbers of quadratic field?
Let $K_1=\Bbb Q(\sqrt{d_1})$ , $K_2=\Bbb Q(\sqrt{d_2})$ and $K=\Bbb Q(\sqrt{d_1},\sqrt{d_2})$.Suppose $h_1,h_2,h$ be class number of $K_1,K_2,K$ respectively.
(i) Can we express $h$ in terms of $...
- 753
4
votes
1
answer
334
views
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.$$...
- 895
2
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0
answers
101
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Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)
Denote the complete elliptic integral of the first kind by
$$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}$$
Question:
Given a ...
- 121
3
votes
1
answer
198
views
Centralizer of the absolute Galois group of a number field
By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
- 4,428
1
vote
0
answers
52
views
Are integration over restricted direct products only useful for specific functions?
So I've been reading Tate's thesis currently. In that we have defined integration of functions on $G$, which are basically formed from restricted direct products of locally compact groups $G_{\...
- 133
13
votes
1
answer
1k
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Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?
I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:
Lemma: if L/K is an abelian ...
- 4,279
4
votes
0
answers
305
views
Explicit invariant map in local class field theory
Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...
- 953
47
votes
1
answer
3k
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A three-line proof of global class field theory?
There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern ...
- 7,972
4
votes
0
answers
347
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Reference request for Kummer-Artin-Schreier-Witt theory
I cannot find the following 4 papers by Sekiguchi–Suwa in their works on Kummer–Artin–Schreier–Witt theory:
On the unified Kummer–Artin–Schreier–Witt theory, Prépublications du laboratoire de ...
- 333
23
votes
1
answer
2k
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Any open Langlands Conjectures for GL_1?
Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?
- 1,629
1
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0
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Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$
Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
7
votes
1
answer
301
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$p$-torsion of class groups
Let $p$ be a fixed odd prime and $\ell$ be another prime such that $\ell \equiv 1 \pmod{p}$.
Consider the number field $\mathbb{Q}(\zeta_p)$ and its extension $\mathbb{Q}(\zeta_p, \zeta_\ell)$. Note ...
- 1,283
3
votes
0
answers
187
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Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
17
votes
3
answers
3k
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L'un des problèmes fondamentaux de la théorie des nombres
In his 1951 report Sur la théorie du corps de classes, Weil writes that
La recherche d'une interprétation de $C_k$ si $k$ est un corps de
nombres, analogue en quelque manière
à l'interprétation ...
- 18.7k
2
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0
answers
110
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Is there data base of quadratic fields which have abelian Hilbert class field?
I did not find it in LMFDB. However, I am looking for a Database; please feel free to navigate me to any paper, which might be helpful.
I really appreciate any help you can provide.
- 343
2
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0
answers
153
views
Construction of genus class fields
Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...
- 577
4
votes
1
answer
304
views
Topological structure on higher dimensional local fields
Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something ...
- 479
3
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0
answers
224
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Central division algebras over $ \mathbb{Q} $
Quaternions over $ \mathbb{Q} $ are an example of a Central Division algebra over $\mathbb{Q} $ for which the basis elements $\{ i,j,ij \} $ other than $1$ are represented by skew-symmetric matrices ...
- 923
2
votes
1
answer
184
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Does K and its Hilbert class field have same conductor?
Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K.$
Definition: (conductor of a abelian number field) Let $K$ be a number field with the abelian Galois group over $\Bbb{Q}....
- 343
3
votes
0
answers
166
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Are class numbers of number fields with prime degree often $1$?
I have taken a look at the class number statistics of the L-functions and Modular Forms Database:
https://www.lmfdb.org/NumberField/stats, table "Distribution by class number".
It appears ...
- 125
2
votes
1
answer
154
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Are the maximal cyclotomic field contained in a number field and its Hilbert class group the same?
Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(...
- 753
15
votes
1
answer
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comparison of completion and Henselization in class field theory
Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
- 1,338
2
votes
2
answers
415
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Computing the class group of a quadratic function field
I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
- 3,362
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3
answers
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Maximal (non-abelian) extensions of number fields unramified everywhere
Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general ...
- 1,269
0
votes
1
answer
192
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English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"
I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
- 577
4
votes
1
answer
418
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Henselian valued fields for characteristic $0$: a characterization
Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property:
$...
- 6,006
16
votes
2
answers
1k
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Central simple algebras approach to class field theory, merits of
As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly ...
- 7,442
3
votes
2
answers
500
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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
15
votes
3
answers
2k
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Ideal classes fixed by the Galois group
Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...
- 1,021
1
vote
1
answer
138
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$0$-th Galois cohomology with topological Milnor K-groups coefficients
In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...
- 479
2
votes
0
answers
294
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Totally ramified extensions of p-adic fields
Let $\mathbb{Q}_p$ denote the field of p-adic numbers. For a prime number $q$ ($\neq p$), does exist a totally ramified extension $K/\mathbb{Q}_p$ with Galois group isomorphis to $\mathbb{Z}_q \times \...
- 437