On numbers which are differences of two conjugates of an algebraic integer

Abstract

We investigate which numbers are expressible as differences of two conjugate algebraic integers. Our first main result shows that a cubic, whose minimal polynomial over the field of rational numbers has the form x3 + px + q, can be written in such a way if p is divisible by 9. We also prove that every root of an integer is a difference of two conjugate algebraic integers, and, more generally, so is every algebraic integer whose minimal polynomial is of the form f (xe) with an integer e ≥ 2.