The other day on this site I came across a question about $\mathbb Z_3 [x]/ \langle x^2 + 1 \rangle$.
At the time I misread the question and thought it was
Find all the irreducible polynomials in the field $\mathbb Z_3 [x]/ \langle x^2 + 1 \rangle$
I was about to post an answer saying that since there are only nine elements, 3 of which are constants, one can determine which elements are irreducible by computing all possible products of the non-constant elements.
As I was typing I realised that this is a stupid answer as, of course, anyone would figure to use brute force to solve the problem and it's not insightful.
Then I recalled that for polynomials over a field of degree 2 and 3 the polynomial is reducible if and only if it has a root in the field.
Unfortunately, I don't know if this can be modified to also apply to equivalence classes of polynomials.
So my question is:
What methods are there to test elements for irreducibility?
Can this theorem about polynomials be somehow adapted to be also applicable to equivalence classes of polynomials?