extensions without unramified subextensions and class number divisibility
Theorem 1.
Let be an extension of number fields such that for any intermediate Galois extension , with , there is at least one finite place or infinite place which ramifies in the extension . Then, , the class number of , divides the class number of , .
First, we deduce some immediate corollaries.
Corollary 1.
Let be an extension of number fields which is totally ramified at some prime (or at an archimedean place). Then divides .
Proof.
The proof is clear since there cannot be unramified subextensions. The theorem applies. ∎
Corollary 2.
Let be a Galois extension of number fields such that is a non-abelian simple group. Then divides .
Proof.
In this case, there cannot be subextensions with abelian Galois group and the theorem applies. ∎
Proof of the Theorem.
Let be the Hilbert class field of . By definition, is the maximal unramified abelian extension of , is isomorphic to , the ideal class group of and . Since there are no nontrivial unramified abelian subextensions of , we have and so . One can show that the extension is unramified and abelian (in fact ). Therefore is contained in , the Hilbert class field of . Hence:
and so, divides . ∎
Title | extensions without unramified subextensions and class number divisibility |
Canonical name | ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility |
Date of creation | 2013-03-22 15:07:35 |
Last modified on | 2013-03-22 15:07:35 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11R29 |
Classification | msc 11R32 |
Classification | msc 11R37 |
Related topic | PushDownTheoremOnClassNumbers |
Related topic | ClassNumberDivisibilityInExtensions |
Related topic | IdealClass |
Related topic | ExistenceOfHilbertClassField |
Related topic | CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois |
Related topic | DecompositionGroup |