# Pisot, Salem and Cyclotomic Polynomials

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Consider the polynomial . The first coefficient is 1, making this, by definition, a monic polynomial. Also by definition, the roots of a monic polynomial are called algebraic integers. The polynomial is irreducible, meaning that it cannot be factored into a product of polynomials with integer coefficients. The highest power is 3, so by the fundamental theorem of algebra, the equation has exactly three complex roots, counting possible multiplicities. Some irreducible monic polynomials with integer coefficients have special names when their roots have certain properties.

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Contributed by: Ed Pegg Jr (October 2013)

Open content licensed under CC BY-NC-SA

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Reference

[1] M. Mossinghoff. "Small Salem Numbers." (Sep 24, 2018) www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html.

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