On the degree of a linear form in conjugates of an algebraic number

Abstract

We investigate the connection between the degree of an algebraic number over a field of characteristic zero and the degree of a linear form in its conjugates. Special attention is given to the case of linear forms in two distinct conjugates. In the process, we show that certain relations with dominant term are impossible, generalizing a result obtained by Smyth for the field of rational numbers. We also prove analogous multiplicative results. As an application, we describe algebraic numbers of prime degree which can be expressed as sums of two distinct conjugates of an algebraic number of the same degree.