extensions without unramified subextensions and class number divisibility

Theorem 1.

Let F/K be an extensionPlanetmathPlanetmath of number fieldsMathworldPlanetmath such that for any intermediate Galois extensionMathworldPlanetmath L/K, with KLF, there is at least one finite place or infinite place which ramifies in the extension L/K. Then, hK, the class numberMathworldPlanetmathPlanetmath of K, divides the class number of F, hF.

First, we deduce some immediate corollaries.

Corollary 1.

Let F/K be an extension of number fields which is totally ramified at some prime (or at an archimedean place). Then hK divides hF.


The proof is clear since there cannot be unramified subextensions. The theorem applies. ∎

Corollary 2.

Let F/K be a Galois extension of number fields such that Gal(F/K) is a non-abelian simple groupMathworldPlanetmathPlanetmath. Then hK divides hF.


In this case, there cannot be subextensions with abelianMathworldPlanetmath Galois groupMathworldPlanetmath and the theorem applies. ∎

Proof of the Theorem.

Let H be the Hilbert class fieldMathworldPlanetmath of K. By definition, H is the maximal unramified abelian extensionMathworldPlanetmathPlanetmath of K, Gal(H/K) is isomorphicPlanetmathPlanetmathPlanetmath to Cl(K), the ideal class group of K and [H:K]=hK. Since there are no nontrivial unramified abelian subextensions of F/K, we have FH=K and so [FH:F]=[H:K]=hK. One can show that the extension FH/F is unramified and abelian (in fact Gal(FH/F)Gal(H/K)). Therefore FH is contained in L, the Hilbert class field of F. Hence:


and so, hK divides hF. ∎

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