Find prime fields over which a polynomial has roots.

Question

Suppose we have a polynomial

$$h(x) = a_n x^n + \dots + a_1 x + 1$$

Given the values $a_1,\ldots,a_n$, how to determine whether there exists such prime $p$ that $h(x)$ has roots over the field $\mathbb{Z}_p = \mbox{GL}(p)$?

Answer 1

Assume $a_n \ne 0, n\ge 1$, let's choose integer $x_0$ large enough so that $h(x_0)$ is a large integer too. Choose a prime factor p of $h(x_0)$, then $x_0$ is root of h(x) over $\mathbb{Z}_p$