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". 



Cyclotomic polynomials.
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.)


