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 $\mathrm{[}K\mathrm{:}\mathrm{Q}\mathrm{]}\mathrm{\le }N$.