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

Abstract

Let $F$ be a field and let $E$ be an étale algebra over $F$, that is, a finite product of finite separable field extensions $E=F_1\times \cdots \times F_r$. 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 $r⩾1$, provided that $F$ is infinite. However, if $F$ is a finite field and $r⩾2$, 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.

12 pages