# unramified extensions and class number divisibility

The following is a corollary of the existence of the Hilbert class field^{}.

###### Corollary 1.

Let $K$ be a number field^{}, ${h}_{K}$ is its class number^{} and let $p$ be a prime. Then $K$ has an everywhere unramified Galois extension^{} of degree $p$ if and only if ${h}_{K}$ is divisible by $p$.

###### Proof.

Let $K$ be a number field and let $H$ be the Hilbert class field of $K$. Then:

$$|\mathrm{Gal}(H/K)|=[H:K]={h}_{K}.$$ |

Let $p$ be a prime number^{}. Suppose that there exists a Galois extension $F/K$, such that $[F:K]=p$ and $F/K$ is everywhere unramified. Notice that any Galois extension of prime degree is abelian^{} (because any group of prime degree $p$ is abelian, isomorphic^{} to $\mathbb{Z}/p\mathbb{Z}$). Since $H$ is the maximal abelian unramified extension^{} of $K$ the following inclusions occur:

$$K\u228aF\subseteq H$$ |

Moreover,

$${h}_{K}=[H:K]=[H:F]\cdot [F:K]=[H:F]\cdot p.$$ |

Therefore $p$ divides ${h}_{K}$.

Next we prove the remaining direction. Suppose that $p$ divides ${h}_{K}=|\mathrm{Gal}(H/K)|$. Since $G=\mathrm{Gal}(H/K)$ is an abelian group (isomorphic to the class group of $K$) there exists a normal subgroup^{} $J$ of $G$ such that $|G/J|=p$. Let $F={H}^{J}$ be the fixed field by the subgroup^{} $J$, which is, by the main theorem of Galois theory^{}, a Galois extension of $K$. This field satisfies $[F:K]=p$ and, since $F$ is included in $H$, the extension $F/K$ is abelian and everywhere unramified, as claimed.
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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 |