Let F/E be a finite Galois extension of fields with abelian Galois group
Γ. A self-dual normal basis for F/E is a normal basis with the
additional property that TrF/E(g(x),h(x))=δg,h for g,h∈Γ.
Bayer-Fluckiger and Lenstra have shown that when char(E)=2, then F
admits a self-dual normal basis if and only if [F:E] is odd. If F/E is an
extension of finite fields and char(E)=2, then F admits a self-dual normal
basis if and only if the exponent of Γ is not divisible by 4. In this
paper we construct self-dual normal basis generators for finite extensions of
finite fields whenever they exist.
Now let K be a finite extension of \Q_p, let L/K be a finite abelian
Galois extension of odd degree and let \bo_L be the valuation ring of L. We
define AL/K to be the unique fractional \bo_L-ideal with square equal to
the inverse different of L/K. It is known that a self-dual integral normal
basis exists for AL/K if and only if L/K is weakly ramified. Assuming
p=2, we construct such bases whenever they exist