# Questions tagged [class-field-theory]

The tag has no usage guidance.

47 questions
Filter by
Sorted by
Tagged with
9answers
13k views

### Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
2answers
6k views

### 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). ...
10answers
15k views

### Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...
17answers
10k views

### Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
8answers
11k views

### Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
4answers
3k views

### Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?

How does one prove that the splitting of primes in a non-abelian extension of number fields is not determined by congruence conditions?
9answers
12k views

### Suggestions for good books on class field theory

Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...
1answer
4k 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 ...
4answers
2k views

### $A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
2answers
764 views

### 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 ...
0answers
322 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 ...
2answers
9k views

### Please check my 6-line proof of Fermat's Last Theorem.

Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small. Here's a result of Eichler (remark after Theorem 6.23 in ...
5answers
7k views

### A reference for geometric class field theory?

The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
3answers
4k views

### 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 ...
1answer
2k views

### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
13answers
6k 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 ...
5answers
2k views

### 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 ...
3answers
2k views

### 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 ...
1answer
1k views

5answers
3k 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.
1answer
1k views

### Can one prove complex multiplication without assuming CFT?

The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
3answers
1k views

### 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 ...
2answers
990 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 $p$ ...
1answer
1k views

### 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 $n=1$?
1answer
876 views