Cyclotomic polynomial

English

Noun

1. Template:Lb For a positive integer n, a polynomial whose roots are the primitive n^th roots of unity, so that its degree is Euler's totient function of n. That is, letting ${\zeta }_n=e^{i2\pi /n}$ be the first primitive n^th root of unity, then ${\mathrm{\Phi }}_n(x)=\prod _{\stackrel{1\le m<n}{\gcd (n,m)=1}}(x-{\zeta }_n^m)$ is the n^th such polynomial. Template:C

   For a prime number $p$, the $p$^th cyclotomic polynomial is $\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+...+x^2+x+1$.

   Cyclotomic polynomials can be shown to be irreducible through the Eisenstein irreducibility criterion, after replacing $x$ with $x+1$.