TITLE: Noncoprime polynomial values
QUESTION [10 upvotes]: Let $p_1, \ldots, p_n$ be a finite sequence of nonconstant polynomials with integer coefficients. Does there exist a finite sequence of integers $x_1, \ldots, x_n$ such that the integers $p_1(x_1), \ldots, p_n(x_n)$ have a nontrivial common divisor?

REPLY [8 votes]: Here is a bit more pedestrian argument. We may assume that $p_i$ are irreducible. Take $\alpha_i$ a root of $p_i$; then $\mathbb Q[\alpha_1,\dots,\alpha_n]=\mathbb Q[\xi]$ by the primitive element theorem. Taking $q$ to be the monic minimal polynomial of $\xi$, we have $\alpha_i=r_i(\xi)$, so $q(x)\mid p_i(r_i(x))$. Now we may take any prime divisor $p$ of the values of $q$ which does not appear in the denominators of the coefficients of $q$ and $r_i$; $p$ divides appropriate values of $p_i$ as well.