Galois Extensions, Finite Fields

Finite Fields

As you recall, the automorphisms of a finite field form a cyclic group. In fact this cycle has length n, when the field has order pⁿ, and the dimension of this field, over Z_p, is also equal to n. The dimension equals the size of the group, hence the extension is galois. Furthermore, the field of order pⁿ is galois over every intermediate extension. Therefore every finite extension of every finite field is galois.

Let F be the algebraic closure of Z_p. Since F has all the roots of everything, it is normal over Z_p. Let u be any element in F. Since Z_p(u) is galois, some automorphism moves u. This automorphism extends to all of F. Thus an automorphism of F moves u, and F is galois over Z_p. The same proof shows F is galois over every finite field.