Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

TL;DR

This paper constructs self-dual normal bases for abelian Galois extensions of finite and local fields, providing explicit methods and conditions for their existence based on the extension properties.

Contribution

It offers explicit constructions of self-dual normal bases in finite and local field extensions, extending known existence results to practical construction methods.

Findings

Self-dual normal bases exist when extension degree is odd or exponent conditions are met.

Constructed explicit generators for finite field extensions with self-dual bases.

Established existence of self-dual integral normal bases in weakly ramified local extensions.

Abstract

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.