salem number

In mathematics, a real 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 Raphal 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 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

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

(

\approx 1.178

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