# 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 $S=\{p_{1},\ldots,p_{r}\}$ be a set of rational primes $p_{i}\in\mathbb{Z}$. 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 $p\in S$. In other words, the only primes that divide the discriminant of $K$ are elements of $S$.

###### Corollary (Hermite’s Theorem).

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

Title Hermite’s theorem HermitesTheorem 2013-03-22 15:05:35 2013-03-22 15:05:35 alozano (2414) alozano (2414) 5 alozano (2414) Corollary msc 11R29 msc 11H06 unramified outside a set of primes