On the number of generators of a separable algebra over a finite field

TL;DR

This paper investigates the minimal number of generators needed for separable algebras over finite fields, providing explicit formulas and bounds, especially highlighting differences from the classical primitive element theorem.

Contribution

It derives a formula for the minimal number of generators of separable algebras over finite fields and establishes bounds for non-commutative cases, extending classical results.

Findings

Provides a formula for the minimal number of generators over finite fields.

Establishes upper and lower bounds for generators of separable algebras.

Highlights the failure of the primitive element theorem for multiple components over finite fields.

Abstract

Let $F$ be a field and let $E$ be an \'etale algebra over $F$, that is, a finite product of finite separable field extensions $E = F_1 \times \dots \times F_r$. The classical primitive element theorem asserts that if $r = 1$, then $E$ is generated by one element as an $F$-algebra. The same is true for any $r \geqslant 1$, provided that $F$ is infinite. However, if $F$ is a finite field and $r \geqslant 2$, the primitive element theorem fails in general. In this paper we give a formula for the minimal number of generators of $E$ when $F$ is finite. We also obtain upper and lower bounds on the number of generators of a (not necessarily commutative) separable algebra over a finite field.