A field theoretic proof of Hermite's theorem for function fields

Let be a finite field. The function field analog of Hermite's theorem says that there are at most finitely many finite separable extensions of inside a fixed separable closure of whose discriminant divisors have bounded degree. In this paper we give a field theoretic proof of this result, inspired by a lemma of Faltings for comparing semisimple -adic Galois representations.