Monogenous algebras. Back to Kronecker

Abstract

In this note we develop some properties of those algebras (called here locally simple) which can be generated by a single element after, if need be, a faithfully flat extension. For finite algebras, this is shown to be in fact a property of the geometric fibers. Morphisms between rings of algebraic integers are locally simple. Expanding an idea introduced by Kronecker we show that much of the properties (in particular local simplicity) of a finite and locally free A-algebra B can be read through the characteristic polynomial of the generic element of B.