Salem number

In mathematics, a real algebraic integer $α > 1$ is a Salem number if all its conjugate roots have absolute value no greater than $1$ , and at least one has absolute value exactly $1$ . Salem numbers are of interest in diophantine approximation and harmonic analysis. They are named for Raphaël Salem (1898-1963).

It can be shown that all the conjugate roots of a Salem number $α$ distinct from $α$ have absolute value exactly one, except one which has absolute value $1 / | α |$ . As a consequence it must be a unit in the ring of algebraic integers, being of norm 1. Because it has a root of absolute value 1, the minimal polynomial for a Salem Number must be reciprocal.