Lehmer's polynomial

In mathematics, a Salem number is a real algebraic integer α > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem.

Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be reciprocal. This implies that 1/α is also a root, and that all other roots have absolute value exactly one. As a consequence α must be a unit in the ring of algebraic integers, being of norm 1.

Every Salem number is a Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).

The smallest known Salem number is the largest real root of Lehmer's polynomial (named after Derrick Henry Lehmer)

which is about x = 1.17628: it is conjectured that it is indeed the smallest Salem number, and the smallest possible Mahler measure of an irreducible non-cyclotomic polynomial.

Lehmer's polynomial is a factor of the shorter 12th-degree polynomial,

all twelve roots of which satisfy the relation

Salem numbers can be constructed from Pisot–Vijayaraghavan numbers. To recall, the smallest of the latter is the unique real root of the cubic polynomial,

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