cyclotomic polynomial

Noun

 * 1)  For a positive integer n, a polynomial whose roots are the primitive nth roots of unity, so that its degree is Euler's totient function of n. That is, letting $$\zeta_n = e^{i 2 \pi / n}$$ be the first primitive nth root of unity, then $$\Phi_n(x) = \prod_\stackrel{1 \le m < n}{\gcd(n,m) = 1} (x - \zeta_n^m) $$ is the nth 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 $$x + 1$$.