cyclotomic polynomial

English

Noun

cyclotomic polynomial (plural cyclotomic polynomials)

1. (algebra) 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 $\Phi _{n}(x)=\prod _{\stackrel {1\leq m<n}{\gcd(n,m)=1}}(x-\zeta _{n}^{m})$ is the n^th such polynomial.

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

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