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?
and
Can this theorem about polynomials be somehow adapted to be also applicable to equivalence classes of polynomials?
Polynomials over the field $\mathbb F_3[x]/\langle x^2 + 1\rangle$ are elements of the polynomial ring $(\mathbb F_3[x]/\langle x^2 + 1\rangle)[z]$. Note that irreducible elements in a ring are by definition nonunits, so the fact that $\mathbb F_3[x]/\langle x^2 + 1\rangle$ is a field makes the question, as you have interpreted it, trivial.