What is the $n$th Cyclotomic Polynomial for a given $n$

Question

Are there any formula to compute the $n$th cyclotomic polynomial for a given $n$? I know how to compute it for small $n$, but for large $n$, I need a general formula..

Answer 1

Here is a general formula: $$ \Phi_{n}(x)=\prod _{d\mid n}(x^{d}-1)^{\mu \left({\frac {n}{d}}\right)} $$ where $\mu$ is the Möbius function. It follows by using Möbius inversion on $$ x^{n}-1=\prod _{d\mid n}\Phi_{d}(x) $$ This also gives a recursive formula for $\Phi _n$: $$ \Phi_n(x) = \frac{x^{n}-1}{\prod _{d\mid n, d\ne n}\Phi_{d}(x)} $$ Note that both formulas express $\Phi_n$ as a rational function which simplifies to a polynomial.

It is unlikely that there is a general formula for the coefficients of $\Phi _n$ in monomial form.