This comment was posted to reddit on Jun 16, 2022 at 5:39 am and was deleted within 8 minutes.

Cyclotomic polynomials are the monic minimal polynomials which contain the roots of unity with coefficient in Z or integers . By minimal polynomial it means that they cannot be reduced into factors in Z that is irreducible . The easiest way to find any nth cyclotomic polynomial is to divide the the polynomial X^n-1 with all the cyclotomic polynomials before n which divides n that is if d|n then divide X^n-1 with all the polynomials of type X^d-1 and all the d must be distinct that is if n=8 then d can be 4 and 2 but since 2 divides 4 the polynomial X^2-1 is already contained in X^4-1 so you dont have to divide by it again . And so for n=8 cyclotomic polynomial is just (X^8-1)/(X^4-1)=X^4+1 . SO try n=9 case yourself and see if you understand the process of calculating them . Its an easy algorithm once we get used to it .