All Questions
Tagged with class-field-theory algebraic-number-theory
89 questions with no upvoted or accepted answers
23
votes
0
answers
1k
views
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 $K$ ...
- 32.6k
21
votes
0
answers
794
views
Class field theory and the class group
Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where $...
- 21.4k
11
votes
1
answer
1k
views
Relationship between the conductor of an order and the conductor of a number field extension
What is the relationship between the conductor $\mathfrak{f}_{\mathcal{o}}$ of an order $\mathcal{o}\subset \mathcal{O}_K$ and the conductor $\mathfrak{f}_{L/K}$ of a field extension in the classical ...
- 693
8
votes
0
answers
2k
views
Characterizing primes that split completely vs. primes with a given splitting behavior
Given a finite abelian extension of number fields $L/K$, the prime ideals $\mathfrak{p}$ in $O_K$ split into primes $\mathfrak{P}$ in $O_L$. The number of primes $\mathfrak{p}$ splits into is ...
- 7,062
7
votes
0
answers
157
views
Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
- 1,116
7
votes
0
answers
205
views
Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$
$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
7
votes
0
answers
128
views
Minimal Discriminants
Let $D_n$ be the minimal absolute value of the discriminants of
number fields with degree $n$. Arnold Scholz conjectured in 1936 that
$D_{397} > D_{400}$, which is, of course, still open (Scholz ...
- 32.6k
6
votes
0
answers
139
views
$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?
Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...
- 363
6
votes
0
answers
737
views
What are the fastest ways to calculate class number of number fields?
Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$?
I am aware that the question is broad but any argument would be helpful.
Some basic approaches I know:...
- 161
6
votes
0
answers
293
views
Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module
Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...
- 1,066
6
votes
0
answers
224
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On a theorem of Dwork and totally ramified extensions
Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal ...
- 61
6
votes
0
answers
221
views
Furtwangler's Principal ideal theorem in number fields
Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes?
The simplest ...
- 61
5
votes
0
answers
176
views
Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
- 528
5
votes
0
answers
127
views
non $p$ part of the class group and analogous results
Washington had proven in 1978 that for $q$ a prime ($q \neq p$), if $q^{e_n}$ exactly divides the class number of $\mathbb{Q}_n$, ie the $n$-th layer in the cyclotomic $\mathbb{Z}_p$ extension, then $...
- 1,283
5
votes
0
answers
121
views
Explicit extensions for Heisenberg groups
Let $G$ be the $p$-adic Heisenberg group $\begin{pmatrix} 1&\mathbb Z_p&\mathbb Z_p\\&1&\mathbb Z_p\\&&1\end{pmatrix}$. Is it possible to write an explicit extension $K/k$, ...
- 621
5
votes
0
answers
758
views
maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
- 51
5
votes
0
answers
196
views
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=...
4
votes
0
answers
160
views
Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields
Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...
- 411
4
votes
0
answers
147
views
A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
- 14.2k
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
4
votes
0
answers
164
views
What are the applications of $\lambda$-rings to class field theory?
In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...
- 804
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
4
votes
0
answers
347
views
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
4
votes
0
answers
506
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
4
votes
0
answers
211
views
Are there "elementary" proofs of the openness of norm subgroups and of the norm limitation theorem?
Let $K$ be a local field and $L/K$ be a finite extension. Let $L^{ab}$ be the maximal abelian subextension of $K$ in $L$. Write $N_L$ (resp. $N_{L^{ab}}$) for the image of the norm map from $L$ (resp. ...
- 1,606
4
votes
0
answers
465
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Reference request: ramified and local geometric class field theory
There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...
- 647
4
votes
0
answers
236
views
Class fields without class field theory
Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...
- 2,375
4
votes
0
answers
190
views
Restricted Iwasawa theory
Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
- 11.3k
4
votes
0
answers
208
views
extending $p$-adic character of the local intertia to the absolute Galois group
Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism
$$
\chi \colon O_{F_v}^\times \to E^\times.
$$
where $O_{F_v}$...
- 1,190
4
votes
0
answers
301
views
Can we prove the uniqueness of the local Artin map by using mostly global class field theory?
Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...
- 6,180
4
votes
0
answers
190
views
Is $K^{ur} K^{\pi} = L$?
Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...
- 6,180
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
3
votes
0
answers
80
views
Local Class field theory and Artin map for the Weil group
I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...
- 367
3
votes
0
answers
215
views
Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
- 85
3
votes
0
answers
162
views
relating class number and narrow class number of a real field
I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
- 637
3
votes
0
answers
224
views
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
3
votes
0
answers
166
views
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
3
votes
0
answers
151
views
Values of Grössencharacter attached to CM elliptic curve
I am trying a cross-post here, as my previous post on stackexchange was not as fruitful as I hoped. The link to the older post is: https://math.stackexchange.com/questions/3327269/values-of-...
- 81
3
votes
0
answers
37
views
augmentation ideal and $\rho$-isotopic spaces
These are two questions regarding Rubin`s paper - Global units and Ideal class groups in Inventiones 1987. The two questions are from section 3.
Let $p$ be a rational prime and $K \subset E \subset F$...
- 151
3
votes
0
answers
107
views
When does a number field have $p$-rank greater than $n$?
Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...
- 1,283
3
votes
0
answers
251
views
The closed subgroup of the idele corresponding to the maximal elementary $p$-extension of a global field
I want to know whether for any global number field $k$, the closed subgroup of the idele corresponding to the maximal elementary $p$-extension ($p$ is a prime number) is $k\ J^p$.
The critical point ...
- 1,013
3
votes
0
answers
203
views
Class field theory for $p$-groups.
I accidentally posted this question to math.stackexchange but think that it is more appropriate here (if not, please say so!):
This question is from Neukirch's book "Algebraic number theory," page ...
- 41
2
votes
0
answers
95
views
Is there a pure algebraic proof of the problem about class number of imaginary quadratic field $p\nmid h_{\mathbb{Q}(\sqrt{-p})}?$
For $p\equiv3\pmod4$, let $h_{\mathbb{Q}(\sqrt{-p})}$ denote the class number of $\mathbb{Q}(\sqrt{-p})$, the class number formula shows $h_{\mathbb{Q}(\sqrt{-p})} < p$. I wonder if there is a pure ...
- 21
2
votes
0
answers
116
views
Restriction of the local Artin map on the valuation ring of a local field
Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
- 367
2
votes
0
answers
65
views
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
2
votes
0
answers
79
views
$n$-th root of character on local field
Let $F$ be a non-Archidean local field of characteristic 0, and $\zeta_n$ the set of $n$-th roots of unity in the algebraic closure of $F$. Assume $\zeta_n\subseteq F$. Let $\chi:F^\times\to\mathbb{C}^...
- 833
2
votes
0
answers
132
views
For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
- 411
2
votes
0
answers
250
views
Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
- 319
2
votes
0
answers
93
views
Compositum of field extensions in context of $\mathbb Z_p$ extension
I had asked this question on stackexchange and I think it is better suited for this site.
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the ...
- 121
2
votes
0
answers
115
views
Artin reciprocity for function fields vs number fields
I have a naive question about the difference of Artin reciprocity in the number field versus function field case:
In the number field case, the double quotient we look at is $$K^\times \backslash \...
- 4,418