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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".

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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.)

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    $\begingroup$ Yes, it was a silly question. Thanks! $\endgroup$
    – user6976
    Commented Jan 16, 2018 at 0:24
  • $\begingroup$ I believe though that you can get something like gcd(m,n) with integer coefficients though. Gerhard "May Review Jameson's Notes Again" Paseman, 2018.01.15. $\endgroup$
    – Gerhard Paseman
    Commented Jan 16, 2018 at 0:57
  • $\begingroup$ I posted a followup question here: mathoverflow.net/questions/290826/cyclotomic-polynomials-2 $\endgroup$
    – user6976
    Commented Jan 16, 2018 at 1:55

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