The images of noncommutative polynomials evaluated on the quaternion algebra

Abstract

Let [Formula: see text] be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field [Formula: see text]. Kaplansky conjectured that for any [Formula: see text], the image of [Formula: see text] evaluated on the set [Formula: see text] of [Formula: see text] by [Formula: see text] matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.