CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS

Let F/E be a finite Galois extension of fields with abelian Galois group Gamma. A self-dual normal basis for F/E is a normal basis with the additional property that Tr-F/E(g(x), h(x)) = delta(g,h) for g, h is an element of Gamma. Bayer-Fluckiger and Lenstra have shown that when char(E) not equal 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 Gamma 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 D-L be the valuation ring of L. We define A(L/K) to be the unique fractional D-L-ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A(L/K) if and only if L/K is weakly ramified. Assuming p not equal 2, we construct such bases whenever they exist.