I am studying for my exam in computer algebra. One topic is the factorization of polynomials in $\mathbb{Q}[x]$. With the Lemma of Gauss we can also look for the factors in $\mathbb{Z}[x]$. The algorithm we discussed has the following basic idea: Let $f \in \mathbb{Z}[x]$ be the polynomial we want to factor and $p$ a prime. We calculate the factorization of $f$ in $\mathbb{F}_p[x]$ and try to recover the irreducible factors of $f$ in $\mathbb{Z}[x]$ from the factors in $\mathbb{F}_p[x]$.
Now i am confused why this is working at all. In my book there is stated that we kind of try all possible factor combinations out of the irreducible factors of $f$ in $\mathbb{F}_p[x]$ whether they divide $f$. Strangely enough this works only if the prime $p$ we selected in the beginning, is selected wisely$(p/2 >$ Mignotte-Bound of $f)$. I absolutely have no idea why under these circumstances i can construct the irreducible factors of $f$ in $\mathbb{Z}[x]$ from these in $\mathbb{F}_p[x]$.
Your statement is not right always. Take $ f(x) = x^3 - 1$ have two irreducible factors $ (x-1) $ and $ ( x^2 + x + 1) $ over $\mathbb{Z} $ , but $ ( x^2 + x + 1) $ is not irreducible over $\mathbb{F}_3 $.