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Let be a finite Galois extension of fields with abelian Galois group . A self-dual normal basis for is a normal basis with the additional property that for . Bayer-Fluckiger and Lenstra have shown that when , then admits a self-dual normal basis if and only if is odd. If is an extension of finite fields and , then admits a self-dual normal basis if and only if the exponent of is not divisible by . In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let be a finite extension of , let be a finite abelian Galois extension of odd degree and let be the valuation ring of . We define to be the unique fractional -ideal with square equal to the inverse different of . It is known that a self-dual integral normal basis exists for if and only if is weakly ramified. Assuming , we construct such bases whenever they exist.