Let $P(n)$ be the set of all monic polynomials of degree $n$ with integer coefficients, such that all coefficients have absolute value at most $2^n$.

Given a positive integer $n$ let us define $A(n)$ as the minimal number $m$ such that for any $f \in P(n)$ there exists a field $k$ with at most $m$ elements such that for some $x\in k$ we have $\bar f(x)=0$, where $\bar f$ is the reduction of $f$ modulo the characteristic of $k$.

Q: What is the asymptotic behaviour of $A(n)$? In particular is it true that $\liminf_{n} \frac{A(n)}{n} = 0$ ?