Separable algebras over infinite fields are 2-generated and finitely presented

Abstract

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M 2(R), under an additional assumption that both pairs generate M 2(R) as an R-algebra.