Irreducible polynomial over $\mathbb F_5$

Question

In my book text of Galois theory:

The polynomial $X^5-X-1$ over $\mathbb Z$. In $\mathbb F_5$ is irreducible.

Could you explain why?

Thanks.

Answer 1

This is an example of an Artin-Schreier polynomial. You can find general proofs about their irreduciblity in this thread and this thread.