# Cyclotomic polynomials.

Let $\Phi_m(x)$ and $\Phi_n(x)$ be two different cyclotomic polynomials. Then $\Phi_m(x)$ and $\Phi_n(x)$ are coprime, so there are two polynomials $s(x), t(x)$ with, say, rational coefficients such that $s(x)\Phi_m(x)+t(x)\Phi_n(x)=1$.

Question. Can one find these $s$ and $t$ with integer coefficients?

I think the answer is "yes".

If this were true, then you would prove that $\Phi_m(x)$ and $\Phi_n(x)$ are coprime after reduction modulo $p$, which is far from true. For instance, $\Phi_4(x)=x^2+1$ and $\Phi_2(x)=x+1$ are not coprime modulo $2$. (Even $\Phi_2(x)$ and $\Phi_1(x)$ are not coprime modulo $2$, of course.)