Salem number

In mathematics, a real_number algebraic integer alpha > 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 alpha have absolute value exactly one, except one which has absolute value 1/|alpha|. As a consequence it must be a Unit (ring theory) in the ring of algebraic integers, being of field norm 1. Because it has a root of absolute value 1, the minimal polynomial for a Salem Number must be reciprocal polynomial.

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

:x^{10} + x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1.

(approx 1.176)

See also: Pisot-Vijayaraghavan number, Mahler measure.