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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..

roydiptajit
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  • You can use that the product of the d-th cyclotomic polynomials, where d is a divisor of n is equal to $x^n -1$, and you should get your answer without too much trouble if $n$ is somewhat reasonable. If it is very large, use sage. – Locally unskillful Nov 26 '20 at 09:01
  • Thanks.. I will check – roydiptajit Nov 26 '20 at 09:20

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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.

lhf
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  • All this and more is available at https://en.wikipedia.org/wiki/Cyclotomic_polynomial – lhf Nov 26 '20 at 10:02
  • See also https://math.stackexchange.com/questions/159487/proving-the-möbius-formula-for-cyclotomic-polynomials – lhf Nov 26 '20 at 10:06