Pisot and Salem Numbers

Abstract

There are two very special classes of algebraic integers that arise repeatedly and naturally in this area of study. Recall that an algebraic integer is any root of any monic polynomial with integer coefficients. A real algebraic integer α is a Pisot number if all its conjugate roots have modulus strictly less than 1. A real algebraic integer α is a Salem number if all its conjugate roots have modulus at most 1, and at least one (and hence (see E2) all but one) of the conjugate roots has modulus exactly 1. As is traditional, though somewhat confusing, we denote the class of all Pisot numbers by S and the class of all Salem numbers by T.