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

Graduation date: 2003Let G be a finite group, G₂ be a Sylow 2-subgroup of G, and L/K be a\ud G-Galois extension. We study the trace form qL/K of L/K and the question of\ud existence of a self-dual normal basis. Our main results are as follows:\ud (1) If G₂ is not abelian and K contains certain roots of unity then qL/K is\ud hyperbolic over K.\ud (2) If C has a subgroup of index 2 then L/K has no orthogonal normal basis for\ud any G-Galois extension L/K.\ud (3) If C has even order and C₂ is abelian then L/K does not have an\ud orthogonal normal basis, for some G-Galois extension L/K.\ud We also give an explicit construction of a self-dual normal basis for an odd\ud degree abelian extension L/K, provided K contains certain roots of unity, and\ud study the generalized trace form for an abelian group G