Generic Extensions and Generic Polynomials for Linear Algebraic Groups

TL;DR

This paper constructs explicit generic polynomials for various linear algebraic groups over fields of positive characteristic, including cyclic 2-groups, using cohomological methods applicable to a broad class of groups.

Contribution

It develops new techniques to explicitly construct generic polynomials for connected linear algebraic groups over finite fields, especially for cyclic 2-groups, with optimal parameter count.

Findings

Constructed generic polynomials for unipotent groups and algebraic tori.

Achieved generic polynomials with minimal parameters for cyclic 2-groups.

Contrasts with Lenstra's theorem by providing generic polynomials where previously none existed.

Abstract

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields satisfying certain conditions on cohomology. In particular, we use our techniques to study constructions for unipotent groups, certain algebraic tori, and certain split semisimple groups. An attractive consequence of our work is the construction of generic polynomials in the optimal number of parameters for all cyclic 2-groups over all fields of positive characteristic. This contrasts with a theorem of Lenstra, which states no cyclic 2-group of order $\ge 8$ has a generic polynomial over $\mathbb{Q}$.