unramified extensions and class number divisibility
The following is a corollary of the existence of the Hilbert class field.
Corollary 1.
Let be a number field, is its class number and let be a prime. Then has an everywhere unramified Galois extension of degree if and only if is divisible by .
Proof.
Let be a number field and let be the Hilbert class field of . Then:
Let be a prime number. Suppose that there exists a Galois extension , such that and is everywhere unramified. Notice that any Galois extension of prime degree is abelian (because any group of prime degree is abelian, isomorphic to ). Since is the maximal abelian unramified extension of the following inclusions occur:
Moreover,
Therefore divides .
Next we prove the remaining direction. Suppose that divides . Since is an abelian group (isomorphic to the class group of ) there exists a normal subgroup of such that . Let be the fixed field by the subgroup , which is, by the main theorem of Galois theory, a Galois extension of . This field satisfies and, since is included in , the extension is abelian and everywhere unramified, as claimed. ∎
Title | unramified extensions and class number divisibility |
Canonical name | UnramifiedExtensionsAndClassNumberDivisibility |
Date of creation | 2013-03-22 15:02:59 |
Last modified on | 2013-03-22 15:02:59 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Corollary |
Classification | msc 11R37 |
Classification | msc 11R32 |
Classification | msc 11R29 |
Related topic | IdealClass |
Related topic | PExtension |
Related topic | Ramify |
Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |