# Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields - Mathematics > Number Theory

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

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 theadditional property that $Tr {F-E}gx,hx=\delta {g,h}$ for $g,h\in\Gamma$.Bayer-Fluckiger and Lenstra have shown that when $charE eq 2$, then $F$admits a self-dual normal basis if and only if $F:E$ is odd. If $F-E$ is anextension of finite fields and $charE=2$, then $F$ admits a self-dual normalbasis if and only if the exponent of $\Gamma$ is not divisible by $4$. In thispaper we construct self-dual normal basis generators for finite extensions offinite fields whenever they exist.Now let $K$ be a finite extension of $\Q p$, let $L-K$ be a finite abelianGalois extension of odd degree and let $\bo L$ be the valuation ring of $L$. Wedefine $A {L-K}$ to be the unique fractional $\bo L$-ideal with square equal tothe inverse different of $L-K$. It is known that a self-dual integral normalbasis exists for $A {L-K}$ if and only if $L-K$ is weakly ramified. Assuming$p eq 2$, we construct such bases whenever they exist.

Author: ** Erik Jarl Pickett**

Source: https://arxiv.org/