# Hermite’s theorem

The following is a corollary of Minkowski’s theorem on ideal classes, which is a corollary of Minkowski’s theorem on lattices.

###### Definition.

Let $S\mathrm{=}\mathrm{\{}{p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{p}_{r}\mathrm{\}}$ be a set of rational primes ${p}_{i}\mathrm{\in}\mathrm{Z}$. We say that a number field^{} $K$ is unramified outside $S$ if any prime not in $S$ is unramified in $K$. In other words, if $p$ is ramified in $K$, then $p\mathrm{\in}S$. In other words, the only primes that divide the discriminant^{} of $K$ are elements of $S$.

###### Corollary (Hermite’s Theorem).

Let $S\mathrm{=}\mathrm{\{}{p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{p}_{r}\mathrm{\}}$ be a set of rational primes ${p}_{i}\mathrm{\in}\mathrm{Z}$ and let $N\mathrm{\in}\mathrm{N}$ be arbitrary. There is only a finite number of fields $K$ which are unramified outside $S$ and bounded^{} degree $\mathrm{[}K\mathrm{:}\mathrm{Q}\mathrm{]}\mathrm{\le}N$.

Title | Hermite’s theorem |
---|---|

Canonical name | HermitesTheorem |

Date of creation | 2013-03-22 15:05:35 |

Last modified on | 2013-03-22 15:05:35 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Corollary |

Classification | msc 11R29 |

Classification | msc 11H06 |

Defines | unramified outside a set of primes |