Cyclotomic polynomials of distinct index are distinct.

Question

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 $u_{(j,n)}=e^{\frac{2\pi i j}{n}}$.

Answer 1

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 $\Phi_n\neq\Phi_m$.

Answer 2

$\Phi_n$ and $\Phi_m$ are different for different $m$ and $n$, because they are coprime polynomials in $\Bbb Q[x]$ for $m\neq n$. Note that it is not enough to consider their degree, because we could have $\phi(m)=\phi(n)$ for different $m$ and $n$, e.g., $\phi(6)=\phi(3)=2$. However, $\Phi_6(x)=x^2-x+1$ and $\Phi_3(x)=x^2+x+1$.

Reference: When are cyclotomic polynomials coprime over the integers?