Irreducible polynomials .

Question

Prove whether the polynomials $f=X^2+ X +3$ and $g=X^3+X+1$ are irreducible in the field $\Bbb Z_5$

The fact that the polynomials are given in pairs made me think that I should find their $\gcd$ and if that equals $1$ then they are irreducible, but I think there is more to if as they could still be factorised even if their gcd is $1$.

Answer 1

Hint. This is easier than you think. If a quadratic or cubic is reducible it has a root. That's an easy short check.

Answer 2

Since both of them have degree smaller than $4$ they are irreducible over $\mathbb Z_5$ if and only if has no root in $\mathbb Z_5$. It turns out that the first polynomial is reducible (it is equal to $(X-1)(X-3)$), whereas the second one is irreducible.