On irreducible algebras spanned by triangularizable matrices

We present several extensions of Burnside’s well-known theorem which states that the only irreducible subalgebra of Mn(F) with algebraically closed field F is Mn(F) itself. We show, among some stronger results, that if F is quasi-algebraically closed (in particular, if F is finite), then the only irreducible subalgebra of Mn(F) that contains a linear basis of triangularizable matrices (a hypothesis that automatically holds in the classical case) is Mn(F) itself. We also consider the problem of “field of definition” for a semigroup S in Mn(K): If a linear functional on S takes values in a smaller field F, is S simultaneously similar to a semigroup in Mn(F)?