cyclotomic polynomial

root of unity

Definition

For any positive integer $n$,the $n$-th cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ is defined as

$${\mathrm{\Phi }}_n(x)=\prod _{\zeta }(x-\zeta ),$$

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

Examples

The first few cyclotomic polynomials are as follows:

${\mathrm{\Phi }}_1(x)$$=x-1$

${\mathrm{\Phi }}_2(x)$$=x+1$

${\mathrm{\Phi }}_3(x)$$=x^2+x+1$

${\mathrm{\Phi }}_4(x)$$=x^2+1$

${\mathrm{\Phi }}_5(x)$$=x^4+x^3+x^2+x+1$

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

${\mathrm{\Phi }}_7(x)$$=x^6+x^5+x^4+x^3+x^2+x+1$

${\mathrm{\Phi }}_8(x)$$=x^4+1$

${\mathrm{\Phi }}_9(x)$$=x^6+x^3+1$

${\mathrm{\Phi }}_{10}(x)$$=x^4-x^3+x^2-x+1$

${\mathrm{\Phi }}_{11}(x)$$=x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$

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

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

${\mathrm{\Phi }}_{105}(x)=$$x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}+x^{34}$

$+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}+x^{15}$

$+x^{14}+x^{13}+x^{12}-x^9-x^8-2x^7-x^6-x^5+x^2+x+1$

Properties

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