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 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.

The smallest known Salem number is the largest real root of the polynomial

x¹⁰ + x⁹ - x⁷ - x⁶ - x⁵ - x⁴ - x³ + x + 1.

()

See also: Pisot-Vijayaraghavan number, Mahler measure.