Let $p$ be a prime number. Prove that there exists a monic polynomial $f\in \mathbf{Z}[x]$ of degree $p$ such that
- for all primes $q\leq p$, the reduction $\overline{f}\in \mathbf{F}_q[x]$ is irreducible.
I am looking for a hint. It may use abstract algebra, Galois theory..
Most of the time I can manage finite fields pretty well, but here I am clueless. I know that any monic irreducible polynomial of degree $p$ in $\mathbf{F}_q[x]$ is a divisor of $X^{q^p}-X$ in $\mathbf{F}_q[x]$. I also thought of taking an Artin-Schreier polynomial $X^p-X+a\in \mathbf{F}_p[x]$ to begin with an irreducible polynomial of degree $p$ in $\mathbf{F}_p[X]$.
Thanks in advance.