Trace forms and self-dual normal bases in Galois field extensions

Let G be a finite group, G ₂ be a Sylow 2-subgroup of G, and L/K be a G-Galois extension. We study the trace form q_L/K of L/K and the question of existence of a self-dual normal basis. Our main results are as follows: (1) If G₂ is not abelian and K contains certain roots of unity then q_L/K is hyperbolic over K. (2) If G has a subgroup of index 2 then L/K has no orthogonal normal basis for any G-Galois extension L/K. (3) If G has even order and G₂ is abelian then L/K does not have an orthogonal normal basis, for some G-Galois extension L/K.; We also give an explicit construction of a self-dual normal basis for an odd degree abelian extension L/K, provided K contains certain roots of unity, and study the generalized trace form for an abelian group G.