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={p1,…,pr} be a set of rational primes pi∈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∈S. In other words, the only primes that divide the discriminant of K are elements of S.
Corollary (Hermite’s Theorem).
Let S={p1,…,pr} be a set of rational primes pi∈Z and let N∈N be arbitrary. There is only a finite number of fields K which are unramified outside S and bounded degree [K:Q]≤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 |