Reduction of polynomials mod p 1-Let $f_1,f_2\in\mathbb{Z}[X]$ be two different irreducible monic polynomials. Is it true that for almost all primes $p$ (that is, for all but a finite number of primes), the polynomials $\bar{f}_1$ and $\bar{f}_2$ have no common roots in $\mathbb{F}_p$?  (here $\bar{f}$ means reduction modulo $p$).  

By Gauss' lemma, $f_1$ and $f_2$ remain irreducible over $\mathbb{Q}[x]$.  The Euclidean algorithm then gives polynomials $g_1,g_2\in\mathbb{Q}[x]$ such that $g_1(x)f_1(x)+g_2(x)f_2(x)=1$.  For all primes $p$ which do not divide the denominator of any coefficient of $g_1$ or $g_2$, we can reduce the above equation mod $p$, and conclude that $\overline{f}_1$ and $\overline{f}_2$ are coprime, so don't share a root.
