examples of cyclotomic polynomials

In this entry we calculate a number of cyclotomic polynomialsMathworldPlanetmath, Φd(x)[x], for various d1. The interested reader can find specific examples at the bottom of the entry. We will concentrate first on the theory details which allow us to calculate these polynomials.

1 The theory behind the examples

The following simple lemma is also useful when calculating cyclotomic polynomials:

Lemma 1.

Let n,d2 be positive integers. If d divides n then Φd(x) divides qn(x)=xn-1x-1.


Let ζd be a primitive dth root of unityMathworldPlanetmath and let Φd(x) be the dth cyclotomic polynomial (which is the minimal polynomial of ζd over ). If d divides n then there is a natural numberMathworldPlanetmath m1 such that n=dm. Thus


and so, ζd is also an nth root of unity and, in particular, it is a root of qn(x). By the properties of minimal polynomials, Φd(x) must divide qn(x). ∎

Lemma 2.

The polynomial Φd(x) is of degree φ(d), where φ is Euler’s phi function.


This is an immediate consequence of the definition: for any positive integer d, we define Φn(x), the dth cyclotomic polynomial, by


where ζd=e2πid, i.e. ζd is an dth root of unity. Therefore, the degree is φ(d). ∎

We begin with the pth cyclotomic polynomials for a prime p2.

Proposition 1.

The polynomial


is irreducible over Q.


In order to show that q(x)=qp(x) is irreducible, we perform a change of variables xx+1, and define q(x)=q(x+1). Clearly, q(x) is irreducible over if and only if q(x) is irreducible. Also:

q(x) = q(x+1)=(x+1)p-1(x+1)-1
= xp+(p1)xp-1+(p2)xp-2++(pp-2)x2+(pp-1)xx
= xp-1+(p1)xp-2+(p2)xp-3++(pp-2)x+(pp-1)

Since all the binomial coefficientsMathworldPlanetmath (pn)=p!(p-n)!n!, for n=1,,p-1, are integers divisible by p, and (pp-1)=p is not divisible by p2, we can use Eisenstein’s criterion to conclude that q(x) is irreducible over . Thus q(x) is irreducible as well, as desired. ∎

As a corollary, we obtain:

Theorem 1.

Let p2 be a prime. Then the pth cyclotomic polynomial is given by


By the lemma, the polynomial Φp(x)[x] divides q(x)=xp-1x-1 and, by the propositionPlanetmathPlanetmath above, q(x) is irreducible. Hence Φp(x)=q(x) as claimed. ∎

The following proposition will be very useful as well:

Proposition 2.

Let n be a positive integer.  Then the binomial  xn-1  has as many prime factorsMathworldPlanetmath (http://planetmath.org/PrimeFactorsOfXn1) with integer coefficients as the integer n has positive divisorsMathworldPlanetmathPlanetmath, both numbers thus being τ(n) (http://planetmath.org/TauFunction).


A proof can be found in this entry (http://planetmath.org/FactorsOfNAndXn1). ∎

2 The examples

A generous list of examples can be found in this entry (http://planetmath.org/PrimeFactorsOfXn1). The examples of Φd(x) can be calculated by recursively factoring the polynomials xn-1, for n1, using (a) the fact that Φp(x)=(xp-1)/(x-1) for primes p2 and (b) the polynomial Φd(x) is a divisor of xn-1 if and only if n is a multipleMathworldPlanetmath of d (and Φd(x) appears with multiplicityMathworldPlanetmath one as a factor, because xn-1 does not have repeated roots). Hence, we can calculate:


and deduce


Before factoring x4-1, note that we know that (x-1) divides it, Φ2(x) divides it and x4-1 has as many divisors as τ(4)=3. Therefore Φ4(x)=(x4-1)/((x-1)(x+1))=x2+1.

The polynomial Φ5(x) is x4+x3+x2+x+1 (by the Theorem). In order to calculate Φ6(x) we factor x6-1. Once again, note that 6 has 4 positive divisors, and we already know the following divisors: x-1, Φ2(x), Φ3(x). Hence:


Notice that we knew a priori (by a Lemma above) that the degree of Φ6(x) is in fact φ(6)=2. Similarly, suppose we want to calculate Φ12. This is a polynomial of degree φ(12)=4, and divides x12-1. On the other hand, x12-1 has τ(12)=6 irreducible factors and we already know the factors corresponding to n=1,2,3,4,6. Thus:


Incidentally, we can find an explicit root of Φ12(x) in terms of radicals. The roots are simply given by:

Title examples of cyclotomic polynomials
Canonical name ExamplesOfCyclotomicPolynomials
Date of creation 2013-03-22 17:20:03
Last modified on 2013-03-22 17:20:03
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Example
Classification msc 11R60
Classification msc 11R18
Classification msc 11C08
Synonym calculating cyclotomic polynomials
Related topic PrimeFactorsOfXn1
Related topic RootOfUnity