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

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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.

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