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 $p$).  

The resultant of two polynomials over a field is an element of the field which is zero if and only if the polynomials have a common factor over the field. The resultant of your two polynomials is an integer, which is zero modulo~$p$ for only finitely many primes $p$, unless the integer is zero, in which case the polynomials have a common factor over the integers. But that is ruled out by the hypotheses. 
The resultant can be calculated as the determinant of a certain matrx whose entries are taken from the coefficients of the polynomials --- it's called the Sylvester matrix.  
