Cyclotomic polynomial

In mathematics, the ${\displaystyle n}$-th cyclotomic polynomial, for any positive integer ${\displaystyle n}$, is the unique irreducible polynomial with integer coefficients that is a divisor of ${\displaystyle x^{n}-1}$ and is not a divisor of ${\displaystyle x^{k}-1}$ for any ${\displaystyle k<n}$. Its roots are all ${\displaystyle n}$-th primitive roots of unity ${\displaystyle e^{2i\pi {\frac {k}{n}}}}$, where ${\displaystyle k}$ runs over the positive integers up to ${\displaystyle n}$ and coprime to ${\displaystyle n}$ (where ${\displaystyle i}$ is the imaginary unit). In other words, the ${\displaystyle n}$-th cyclotomic polynomial is equal to

${\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right).}$

It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (${\displaystyle e^{2i\pi /n}}$ is an example of such a root).

An important relation linking cyclotomic polynomials and primitive roots of unity is

${\displaystyle \prod _{d\mid n}\Phi _{d}(x)=x^{n}-1,}$

showing that ${\displaystyle x}$ is a root of ${\displaystyle x^{n}-1}$ if and only if it is a ${\displaystyle d}$-th primitive root of unity for some ${\displaystyle d}$ that divides ${\displaystyle n}$.