All Questions
Tagged with analytic-number-theory class-field-theory
7 questions
4
votes
1
answer
224
views
Generators of the ideal class group
Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
- 43
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
9
votes
2
answers
902
views
How to compute with the Stark conjectures?
I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...
- 7,633
7
votes
1
answer
389
views
Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$
Let $x\in\{\text{totally ramified, inert, totally split}\}.$
If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so ...
- 1,805
15
votes
1
answer
1k
views
Is there an elegant algebraic proof of this formula for quadratic field discriminants?
Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "...
- 7,347
67
votes
17
answers
12k
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 ...
- 7,127
17
votes
6
answers
3k
views
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