# cyclotomic polynomial

## Definition

For any positive integer $n$, the $n$-th $\Phi_{n}(x)$ is defined as

 $\Phi_{n}(x)=\prod_{\zeta}(x-\zeta),$

where $\zeta$ ranges over the primitive $n$-th roots of unity (http://planetmath.org/RootOfUnity).

## Examples

The first few cyclotomic polynomials are as follows:

 $\displaystyle\Phi_{1}(x)$ $\displaystyle=x-1$ $\displaystyle\Phi_{2}(x)$ $\displaystyle=x+1$ $\displaystyle\Phi_{3}(x)$ $\displaystyle=x^{2}+x+1$ $\displaystyle\Phi_{4}(x)$ $\displaystyle=x^{2}+1$ $\displaystyle\Phi_{5}(x)$ $\displaystyle=x^{4}+x^{3}+x^{2}+x+1$ $\displaystyle\Phi_{6}(x)$ $\displaystyle=x^{2}-x+1$ $\displaystyle\Phi_{7}(x)$ $\displaystyle=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ $\displaystyle\Phi_{8}(x)$ $\displaystyle=x^{4}+1$ $\displaystyle\Phi_{9}(x)$ $\displaystyle=x^{6}+x^{3}+1$ $\displaystyle\Phi_{10}(x)$ $\displaystyle=x^{4}-x^{3}+x^{2}-x+1$ $\displaystyle\Phi_{11}(x)$ $\displaystyle=x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ $\displaystyle\Phi_{12}(x)$ $\displaystyle=x^{4}-x^{2}+1$

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

 $\displaystyle\Phi_{105}(x)=$ $\displaystyle x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+% x^{35}+x^{34}$ $\displaystyle\phantom{x^{48}}+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}% -x^{20}+x^{17}+x^{16}+x^{15}$ $\displaystyle\phantom{x^{48}}+x^{14}+x^{13}+x^{12}-x^{9}-x^{8}-2x^{7}-x^{6}-x^% {5}+x^{2}+x+1$

## Properties

For every positive integer $n$, $\Phi_{n}(x)$ is an irreducible polynomial of degree $\phi(n)$ in $\mathbb{Q}[x]$, and is the minimal polynomial of each primitive $n$-th root of unity. Here $\phi(n)$ 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