I am trying to understand Galois Theory and reading through various theorems and lemmas, some of which are still confusing me. A lemma proved by Artin states that if F is the fixed field of a finite group G of automorphisms in a field E, then the degree [E:F] ≤ |G| = n. The proof relies on setting up a linear dependence relationship for any set of n+1 elements in E, using n equations in n+1 unknowns with coefficients obtained by every permutation in G. I am trying to get an intuitive understanding of the result. I already know that in the case where G is the Galois group G(E/F) and E is a splitting field of a separable polynomial over F, then F is the fixed field of G and [E:F] = |G|. But what properties of E and G could cause [E:F] < |G|? I can't find any example where strict inequality arises.