All Questions
49 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 ...
- 7,072
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 ...
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
738
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
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 $$\...
- 528
5
votes
0
answers
758
views
maximal abelian extension of quadratic extension of $\mathbb Q_p$
I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,...
- 51
5
votes
0
answers
196
views
Analogue of a ring extension splitting in the Kummer case
Background (the Kummer extension case)
Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=...
4
votes
0
answers
147
views
A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
- 14.2k
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
4
votes
0
answers
164
views
What are the applications of $\lambda$-rings to class field theory?
In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...
- 804
4
votes
0
answers
305
views
Explicit invariant map in local class field theory
Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...
- 953
4
votes
0
answers
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
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
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
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
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
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
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
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
125
views
Conditions for being an entry in a trace compatible sequence
$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal ...
- 123
2
votes
0
answers
110
views
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
votes
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
2
votes
0
answers
327
views
Definition of Euler system of cyclotomic units
I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt:
Let $F=\mathbb{Q}(\mu_m)$.
Let $\mathcal{I}(m)$ = {positive square free integers divisible only by ...
- 99
2
votes
0
answers
168
views
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
2
votes
0
answers
58
views
generator of ring class field extension
everyone! I have another questions. Let $K=\mathbb{Q}(\sqrt{-3})$ be an imaginary quadratic field and let $p\equiv 8\mod 9$ be a prime. Denote $H_{3p}$ and $H_{p}$ for the ring class field of $K$ with ...
- 390
2
votes
0
answers
169
views
A version of weak approximation with S-integers
Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let
$$K_S = \prod_{v\in S} K_v$$
where $K_v$ is the completion of $K$ at $v$. For $v\in S$, ...
- 323
2
votes
0
answers
100
views
Quasi-algebraically closed fields reference request
I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952.
My current background is the first 6 chapters from ...
- 87
2
votes
0
answers
233
views
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 ...
- 4,593
1
vote
0
answers
58
views
Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
- 14.2k
1
vote
0
answers
100
views
Examples of $\mathbb{Z}_p$-extensions and two $\mathbb{Z}_p$-extensions with a "nontrivial" intersection
Let $k$ be a number field (a finite extension of $\mathbb{Q}$). Let $p$ be a prime. By saying "$\mathbb{Z}_p$-extensions", we mean Galois extensions $K/k$ of Galois group isomorphic to $\...
- 639
1
vote
0
answers
116
views
CM-fields and ideal theoretic ray class group
Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers
and $S$ a finite subset of the real places. Let $\mathfrak{m}
\subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...
- 5,986
1
vote
0
answers
114
views
Rayclass group and Hilbert class group, $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/\mathfrak{a})^\times$
Let $K$ be an imaginary quadratic field with class number $1$. Let $\mathfrak{a}$ be an ideal of $ \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.
Why ...
- 1,541
1
vote
0
answers
200
views
Units in residue classes modulo prime ideal
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...
- 11
1
vote
0
answers
54
views
Finite generation for a restricted ramification idele module
Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...
- 11.3k
1
vote
0
answers
151
views
Skew symmetry for the Hilbert symbol
Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...
- 33
0
votes
0
answers
200
views
What is the conductor of $K(\sqrt{2})$ over $K$?
Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...
- 753
0
votes
0
answers
87
views
Image of extension ideal classes homomorphism in ideal class group under Artin map in class field theory
Let $K/P$ be a finite extension of number fields and $\epsilon_{K/P}:[\mathfrak{a}] \in Cl(P) \rightarrow [\mathfrak{a}.\mathcal{O}_K]\in Cl(K)$ be the ideal class transfer homomorphism. It's well ...
- 437