# extensions without unramified subextensions and class number divisibility

###### Theorem 1.

Let $F/K$ be an extension of number fields such that for any intermediate Galois extension $L/K$, with $K\subsetneq L\subsetneq F$, there is at least one finite place or infinite place which ramifies in the extension $L/K$. Then, $h_{K}$, the class number of $K$, divides the class number of $F$, $h_{F}$.

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 $h_{K}$ divides $h_{F}$.

###### Proof.

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 $\operatorname{Gal}(F/K)$ is a non-abelian simple group. Then $h_{K}$ divides $h_{F}$.

###### Proof.

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

###### Proof of the Theorem.

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

 $h_{F}=[L:F]=[L:FH]\cdot[FH:F]=[L:FH]\cdot[H:K]=[L:FH]\cdot h_{K}$

and so, $h_{K}$ divides $h_{F}$. ∎

 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