Is this polynomial in a finite field irreducible?

Question

Let $\mathbb{F}:= \{a_1, \ldots , a_n \}$ be a finite field. Consider the following polynomial $$f = \prod_{i = 1}^n (X-a_i) + 1 $$ Is this polynomial irreducible? If not, is it irreducible in $\mathbb{F}_p$, a finite field with $p$ Elements for a prime $p$?