Stephen D. Cohen (Department of Mathematics, University of Glasgow, UK)
Existence theorems for generator polynomials over finite fields
4 de Fevereiro de 2009 (quarta-feira), às 14h30m, no Anfiteatro
Abstract
Let $\mathbb{F}_q$ denote the finite field of prime power order $q$ and $\mathbb{F}_{q^n}$ its extension of degree $n$. Generator polynomials over $\mathbb{F}_q$ of degree $n$ include \emph{primitive} polynomials, whose roots each generate $\mathbb{F}_q$ multiplicatively, and \emph{normal} polynomials, when the roots form a normal basis $\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$ of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$. Using character sums and estimates it can be relatively easy to establish asymptotic theorems about the existence of such polynomials with specified properties in the sense that these theorems are valid provided $q$ or $n$ is sufficiently large. The focus of this talk is to describe existence results (and the methods employed to find them) that are complete for all values of $q$ and $n$ for which the relevant question makes sense, with all exceptions listed.