examples of cyclotomic polynomials

In this entry we calculate a number of cyclotomic polynomials, ${\mathrm{\Phi }}_d(x)\in \mathbb{Q}[x]$, for various $d\ge 1$. 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 $n\mathrm{,}d\mathrm{\ge }\mathrm{2}$ be positive integers. If $d$ divides $n$ then ${\mathrm{\Phi }}_d\mathit{}\mathrm{(}x\mathrm{)}$ divides $q_n\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\frac{x^n\mathrm{-}\mathrm{1}}{x\mathrm{-}\mathrm{1}}$.

Proof.

Let ${\zeta }_d$ be a primitive $d$th root of unity and let ${\mathrm{\Phi }}_d(x)$ be the $d$th cyclotomic polynomial (which is the minimal polynomial of ${\zeta }_d$ over $\mathbb{Q}$). If $d$ divides $n$ then there is a natural number $m\ge 1$ such that $n=dm$. Thus

$${\zeta }_d^n={({\zeta }_d^d)}^m=1^m=1$$

and so, ${\zeta }_d$ is also an $n$th root of unity and, in particular, it is a root of $q_n(x)$. By the properties of minimal polynomials, ${\mathrm{\Phi }}_d(x)$ must divide $q_n(x)$. ∎

Lemma 2.

The polynomial ${\mathrm{\Phi }}_d\mathit{}\mathrm{(}x\mathrm{)}$ is of degree $\phi \mathit{}\mathrm{(}d\mathrm{)}$, where $\phi$ is Euler’s phi function.

Proof.

This is an immediate consequence of the definition: for any positive integer $d$, we define ${\mathrm{\Phi }}_n(x)$, the $d$th cyclotomic polynomial, by

$${\mathrm{\Phi }}_d(x)=\prod _{\stackrel{j=1,}{(j,d)=1}}^d(x-\zeta _d{}^j)$$

where ${\zeta }_d=e^{\frac{2\pi i}{d}}$, i.e. ${\zeta }_d$ is an $d$th root of unity. Therefore, the degree is $\phi (d)$. ∎

We begin with the $p$th cyclotomic polynomials for a prime $p\ge 2$.

Proposition 1.

The polynomial

$$q_p(x)=\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+\mathrm{\cdots }+x^2+x+1$$

is irreducible over $\mathrm{Q}$.

Proof.

In order to show that $q(x)=q_p(x)$ is irreducible, we perform a change of variables $x\mapsto x+1$, and define $q^{\prime }(x)=q(x+1)$. Clearly, $q^{\prime }(x)$ is irreducible over $\mathbb{Q}$ if and only if $q(x)$ is irreducible. Also:

$q^{\prime }(x)$ $=$ $q(x+1)={\displaystyle \frac{{(x+1)}^p-1}{(x+1)-1}}$

$=$ ${\displaystyle \frac{x^p+\left(\genfrac{}{}{0pt}{}{p}{1}\right)x^{p-1}+\left(\genfrac{}{}{0pt}{}{p}{2}\right)x^{p-2}+\mathrm{\cdots }+\left(\genfrac{}{}{0pt}{}{p}{p-2}\right)x^2+\left(\genfrac{}{}{0pt}{}{p}{p-1}\right)x}{x}}$

$=$ $x^{p-1}+\left({\displaystyle \genfrac{}{}{0pt}{}{p}{1}}\right)x^{p-2}+\left({\displaystyle \genfrac{}{}{0pt}{}{p}{2}}\right)x^{p-3}+\mathrm{\cdots }+\left({\displaystyle \genfrac{}{}{0pt}{}{p}{p-2}}\right)x+\left({\displaystyle \genfrac{}{}{0pt}{}{p}{p-1}}\right)$

Since all the binomial coefficients $\left(\genfrac{}{}{0pt}{}{p}{n}\right)=\frac{p!}{(p-n)!n!}$, for $n=1,\mathrm{\dots },p-1$, are integers divisible by $p$, and $\left(\genfrac{}{}{0pt}{}{p}{p-1}\right)=p$ is not divisible by $p^2$, we can use Eisenstein’s criterion to conclude that $q^{\prime }(x)$ is irreducible over $\mathbb{Q}$. Thus $q(x)$ is irreducible as well, as desired. ∎

As a corollary, we obtain:

Theorem 1.

Let $p\mathrm{\ge }\mathrm{2}$ be a prime. Then the $p$th cyclotomic polynomial is given by

$${\mathrm{\Phi }}_p(x)=\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+\mathrm{\cdots }+x+1.$$

Proof.

By the lemma, the polynomial ${\mathrm{\Phi }}_p(x)\in \mathbb{Q}[x]$ divides $q(x)=\frac{x^p-1}{x-1}$ and, by the proposition above, $q(x)$ is irreducible. Hence ${\mathrm{\Phi }}_p(x)=q(x)$ as claimed. ∎

The following proposition will be very useful as well:

Proposition 2.

Let $n$ be a positive integer.  Then the binomial  $x^n\mathrm{-}\mathrm{1}$  has as many prime factors (http://planetmath.org/PrimeFactorsOfXn1) with integer coefficients as the integer $n$ has positive divisors, both numbers thus being $\tau \mathit{}\mathrm{(}n\mathrm{)}$ (http://planetmath.org/TauFunction).

Proof.

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 ${\mathrm{\Phi }}_d(x)$ can be calculated by recursively factoring the polynomials $x^n-1$, for $n\ge 1$, using (a) the fact that ${\mathrm{\Phi }}_p(x)=(x^p-1)/(x-1)$ for primes $p\ge 2$ and (b) the polynomial ${\mathrm{\Phi }}_d(x)$ is a divisor of $x^n-1$ if and only if $n$ is a multiple of $d$ (and ${\mathrm{\Phi }}_d(x)$ appears with multiplicity one as a factor, because $x^n-1$ does not have repeated roots). Hence, we can calculate:

$$x-1,x^2-1=(x-1)(x+1),x^3-1=(x-1)(x^2+x+1)$$

and deduce

$${\mathrm{\Phi }}_1(x)=x-1,{\mathrm{\Phi }}_2(x)=x+1,{\mathrm{\Phi }}_3(x)=x^2+x+1.$$

Before factoring $x^4-1$, note that we know that $(x-1)$ divides it, ${\mathrm{\Phi }}_2(x)$ divides it and $x^4-1$ has as many divisors as $\tau (4)=3$. Therefore ${\mathrm{\Phi }}_4(x)=(x^4-1)/((x-1)(x+1))=x^2+1$.

The polynomial ${\mathrm{\Phi }}_5(x)$ is $x^4+x^3+x^2+x+1$ (by the Theorem). In order to calculate ${\mathrm{\Phi }}_6(x)$ we factor $x^6-1$. Once again, note that $6$ has $4$ positive divisors, and we already know the following divisors: $x-1$, ${\mathrm{\Phi }}_2(x)$, ${\mathrm{\Phi }}_3(x)$. Hence:

$${\mathrm{\Phi }}_6(x)=\frac{x^6-1}{(x-1)(x+1)(x^2+x+1)}=x^2-x+1.$$

Notice that we knew a priori (by a Lemma above) that the degree of ${\mathrm{\Phi }}_6(x)$ is in fact $\phi (6)=2$. Similarly, suppose we want to calculate ${\mathrm{\Phi }}_{12}$. This is a polynomial of degree $\phi (12)=4$, and divides $x^{12}-1$. On the other hand, $x^{12}-1$ has $\tau (12)=6$ irreducible factors and we already know the factors corresponding to $n=1,2,3,4,6$. Thus:

$${\mathrm{\Phi }}_{12}(x)=\frac{x^{12}-1}{(x-1)(x+1)(x^2+x+1)(x^2+1)(x^2-x+1)}=x^4-x^2+1.$$

Incidentally, we can find an explicit root of ${\mathrm{\Phi }}_{12}(x)$ in terms of radicals. The roots are simply given by:

$$x^2=\frac{1\pm \sqrt{-3}}{2},x=\pm \sqrt{\frac{1\pm \sqrt{-3}}{2}}.$$

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