On groups and semigroups of matrices with spectra in a finitely generated field

Abstract

Let K be an algebraically closed field of characteristic zero and F<K a subfield which is finitely generated over the prime field. Assume is a semigroup of invertible matrices such that the spectra of all the elements of are contained in F. Then the group , generated by , contains a solvable normal subgroup of finite index. As a consequence, it follows that an irreducible semigroup such that the spectra of all the elements of are contained in F is conjugate to a subsemigroup of Mn (F).