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). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial PK,α(t) of α over K. If K is given inside an algebraically closed field C, then the conjugates can be taken inside... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

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. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 over F of least degree such that p(A)=0. ... In mathematics, for a polynomial p with complex coefficients, we define where denotes the complex conjugate of A polynomial is called reciprocal if p(z) = p*(z). ...

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

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

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See also: Pisot-Vijayaraghavan number, Mahler measure. In mathematics, a Pisot-Vijayaraghavan number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value. ... In mathematics, the Mahler measure of a polynomial p is Here p is assumed complex-valued and is the lα norm of p. ...