If $n$ is a positive integer, the $n$th cyclotomic polynomial is defined to be the unique irreducible polynomial with integer coefficients which is a divisor of $x^n - 1$, but not of $x^k - 1$ for any $0 < k < n$.

Alternatively, the $n$th cyclotomic polynomial can be written as

$$\Phi_{n}(x) = \prod_{\stackrel{1\le k\le n}{\gcd(k,n)=1}} (x - e^{2i\pi \frac{k}{n}})$$

Source: Cyclotomic polynomial.

history | show excerpt | excerpt history