normal basis theorem

Usually we pick a basis of an extension somewhat arbitrarily — like 1, sqrt(2) for Q(sqrt(2)). The normal basis theorem says you can do something far more elegant: choose a single clever element so that it and all of its images under the Galois group together form a basis. The whole basis is then one orbit, shuffled around by the group.

Precisely, if L/K is a finite Galois extension with Galois group G = {sigma_1, ..., sigma_n}, then there exists an element alpha in L whose conjugates sigma_1(alpha), ..., sigma_n(alpha) form a K-basis of L. Such an alpha is called a normal basis generator, and the basis it produces is a normal basis. Equivalently, L is a free module of rank 1 over the group algebra K[G]; the regular representation of G is realized concretely inside L.

This is more than aesthetic. Because G permutes the normal basis exactly as it permutes itself by left multiplication, the action of the Galois group on L becomes the regular representation, which is computationally and theoretically convenient — normal bases are used in fast arithmetic over finite fields and in describing the structure of L as a Galois module. The proof in the infinite-field case uses a Vandermonde/independence-of-characters argument; the finite-field case needs a separate, more careful treatment.