  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 $g,h\in\Gamma$.
Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, 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 $\bo_L$ be the valuation ring of $L$. We
define $A_{L/K}$ 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 $A_{L/K}$ if and only if $L/K$ is weakly ramified. Assuming
$p\neq 2$, we construct such bases whenever they exist.
