Irreducible of $\mathbb Z/p\mathbb Z$. What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no sense, how can I prove using Eisenstein that $X^5+2X+2$ is irreducible over $\mathbb Z/3\mathbb Z$ ?

Let $F_9$ be the field $9$ elements. If the polynomial is reducible, it will have a factor of degree $\leq 2$, hence a root in $F_9$. Therefore it is enough to show that it has no roots in $F_9$. Assume $x$ is such a root. Clearly, $x \ne 0,1$.
The group $F_9^{*}$ has eight elements. So by Lagrange's theorem, $x^8 = 1$, hence $x^4 = \pm 1$. Therefore, we have $0 = x^5 + 2x + 2 = \pm x + 2x + 2.$ But this implies $x = 1$, a contradiction.
