# extensions without unramified subextensions and class number divisibility

###### Theorem 1.

Let $F\mathrm{/}K$ be an extension^{} of number fields^{} such that for any intermediate Galois extension^{} $L\mathrm{/}K$, with $K\mathrm{\u228a}L\mathrm{\u228a}F$, there is at least one finite place or infinite place which ramifies in the extension $L\mathrm{/}K$. 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 $F\mathrm{/}K$ 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 $F\mathrm{/}K$ be a Galois extension of number fields such that $\mathrm{Gal}\mathit{}\mathrm{(}F\mathrm{/}K\mathrm{)}$ 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 $K$, $\mathrm{Gal}(H/K)$ is isomorphic^{} to $\mathrm{Cl}(K)$, 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 $\mathrm{Gal}(FH/F)\cong \mathrm{Gal}(H/K)$). 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 |