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

Abstract

We continue the investigation started by A. Dubickas of the numbers which are differences of two conjugates of an algebraic integer over the field Q of rational numbers. Mainly, we show that the cubic algebraic integers over Q with zero trace satisfy this property and we give a characterisation for those for which this property holds in their normal closure. We also prove that if a normal extension K/Q is of prime degree, then every integer of K with zero trace is a difference of two conjugates of an algebraic integer in K if and only if there exists an integer of K with trace 1.