cyclotomic polynomial


root of unityMathworldPlanetmath


For any positive integer n, the n-th cyclotomic polynomialMathworldPlanetmath Φn(x) is defined as


where ζ ranges over the primitive n-th roots of unity (


The first few cyclotomic polynomials are as follows:

Φ1(x) =x-1
Φ2(x) =x+1
Φ3(x) =x2+x+1
Φ4(x) =x2+1
Φ5(x) =x4+x3+x2+x+1
Φ6(x) =x2-x+1
Φ7(x) =x6+x5+x4+x3+x2+x+1
Φ8(x) =x4+1
Φ9(x) =x6+x3+1
Φ10(x) =x4-x3+x2-x+1
Φ11(x) =x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1
Φ12(x) =x4-x2+1

The preceding examples may give the impression that the coefficientsMathworldPlanetmath are always -1, 0 or 1, but this is not true in general. For example,

Φ105(x)= x48+x47+x46-x43-x42-2x41-x40-x39+x36+x35+x34


For every positive integer n, Φn(x) is an irreducible polynomialMathworldPlanetmath of degree ϕ(n) in [x], and is the minimal polynomial of each primitive n-th root of unity. Here ϕ(n) is Euler’s phi function.

Title cyclotomic polynomial
Canonical name CyclotomicPolynomial
Date of creation 2013-03-22 12:36:00
Last modified on 2013-03-22 12:36:00
Owner yark (2760)
Last modified by yark (2760)
Numerical id 14
Author yark (2760)
Entry type Definition
Classification msc 11R60
Classification msc 11R18
Classification msc 11C08
Related topic AllOnePolynomial
Related topic FactoringAllOnePolynomialsUsingTheGroupingMethod
Related topic CyclotomicField
Related topic RootOfUnity