7
votes
1answer
322 views

Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...
3
votes
1answer
198 views

Unramified extension and class field theory

I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me. Let $F$ be a ...
5
votes
0answers
241 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 ...
4
votes
1answer
330 views

About principal ideal theorem in number fields

I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$. In this case, there is a well-known result that "every ambiguous class in the class group ...
7
votes
1answer
432 views

Numbers integrally represented by a ternary cubic form

Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$ $$ f(a,b,c) = \det \left( \begin{array}{ccc} a & b ...
9
votes
1answer
603 views

The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
6
votes
1answer
352 views

Generalization of Hilbert 94 and capitulation

Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
11
votes
1answer
429 views

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 ...
6
votes
1answer
401 views

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 ...
1
vote
1answer
166 views

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 ...
2
votes
1answer
366 views

class groups of unramified cyclic p-extensions of imaginary quadratic fields

Let $K$ be an imaginary quadratic number field with $p$-Sylow-class group $A(K)$ and $L/K$ be an unramified cyclic extension of $K$ of degree $p$ ($p$ prime). Then I am looking for heuristics on ...
9
votes
0answers
474 views

Class groups in dihedral extensions - some sort of Spiegelungssatz?

Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...
16
votes
1answer
757 views

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 ...
1
vote
0answers
181 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 ...
15
votes
0answers
634 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 $K$ ...
12
votes
1answer
532 views

Capitulation in cyclotomic extensions

Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$ (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of ...
3
votes
0answers
122 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 ...
6
votes
2answers
1k views

Is there a notion of Galois extension for Z / p^2?

The above title is in fact a special case of what I want to ask. Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}_p $. The intersections of these extensions to the ring ...
6
votes
1answer
573 views

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 ...
6
votes
4answers
916 views

Origins of functional field arithmetic

Background: By function field, we mean a finite extension of the field of rational functions of one variable over a finite field with $p$ elements. Classfield theory for function fields was ...
13
votes
5answers
1k views

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.
30
votes
1answer
3k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
40
votes
13answers
4k views

Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details). IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...
10
votes
6answers
1k views

Reference for Learning Global Class Field Theory Using the Original Analytic Proofs?

Hi Everyone! 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 ...