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 the primitive $n$-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 coefficients are 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.

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