# Cyclotomic polynomials

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circular polynomials

The polynomials that satisfy the relation where the product is taken over all positive divisors of the number , including itself. Over the field of complex numbers one has where ranges over the primitive -th roots of unity (cf. Primitive root). The degree of is the number of integers among that are relatively prime with . The polynomials can be computed recursively by dividing the polynomial by the product of all , , . The coefficients lie in the prime field ; in case of characteristic zero, they are integers. Thus, If, moreover, is prime, then The polynomial can be explicitly expressed using the Möbius function : For example,   All the polynomials are irreducible over the field of rational numbers, but they may be reducible over finite prime fields. Thus, the relation is valid over the field of residues modulo 11.

The equation , which gives all primitive -th roots of unity, is known as the equation of division of the circle. The solution of this equation in trigonometric form is where the fraction is irreducible, i.e. and are relatively prime. The solution of the equation of division of the circle by radicals is closely connected with the problem of constructing a regular -gon, or with the equivalent problem of subdividing the circle into equal parts; the latter problem can be solved using a straightedge and a pair of compasses if and only if the equation is solvable in quadratic radicals. It was shown by C.F. Gauss in 1801 that this condition is satisfied if and only if where is a non-negative integer and are pairwise different prime numbers of the form , where is a non-negative integer.

How to Cite This Entry:
Cyclotomic polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclotomic_polynomials&oldid=16160
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article