On primitive element of finite k-algebras and applications to commuting matrices

Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the univariate polynomials f1, ..., fn is inseparable. This result lead to some Corollaries related to the existence of primitive element of finite k-algebras. Finally, these results are further used to investigate the well known Frobenius question, whether two commuting matrices A and B can be expressed as polynomials in some matrix C. More specifically, we identify certain classes of matrices for which matrix C exists and different classes where no such matrix C exists.