The eigenvalue field is a splitting field

Abstract.

Let K be an algebraically closed field of arbitrary characteristic and F < K a subfield. If ${\mathcal{S}} \subset M_{n}(K)$ is an irreducible semigroup of matrices such that the spectra of all the elements of ${\mathcal{S}}$ are contained in F, then ${\mathcal{S}}$ is conjugate to a subsemigroup of M _n (F).