### Abstract:

Let G be a finite group, G₂ be a Sylow 2-subgroup of G, and L/K be a
G-Galois extension. We study the trace form qL/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 qL/K is
hyperbolic over K.
(2) If C has a subgroup of index 2 then L/K has no orthogonal normal basis for
any G-Galois extension L/K.
(3) If C has even order and C₂ 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.