Finite splitting fields of normal subgroups

Abstract

Let G be a finite group, $ N \triangleleft G $ a normal subgroup, p a prime, $ K = \mathbb{F}_{p^{k}} $ a finite splitting field of characteristic p for G and $ n := \exp (G/N). $ We prove that $ L := \mathbb{F}_{p^{kn}} $ is a splitting field for N, using the action of the Galois group of the field extension $ K \subset L $ on the irreducible representations of N. As $ \mathbb{F}_{p} $ is a splitting field for the symmetric group S n we get as a corollary that $ \mathbb{F}_{p^2} $ is a splitting field for the alternating group A n .