Does there exist two cyclotomic polynomial $\Phi_n$ and $\Phi_m$ which are equal but $n\neq m$?
The cyclotomic polynomial is defined as $\Phi_n(x)=\prod_{\substack{1\le j\le n \\ \gcd(j,n)=1}}(x-u_{(j,n)})$ for which $u_{(j,n)}=e^{\frac{2\pi i j}{n}}$.
Best Answer
From your definition it is immediate that if $n\neq m$ then $$\Phi_n(u_{(1,n)})=0 \qquad\text{ and }\qquad \Phi_m(u_{(1,n)})\neq0,$$ which shows that $\Phi_n\neq\Phi_m$.