Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4.$ 
Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4$.  Prove that you have found them all and that the ones you found are irreducible.

I am looking for some sort of way to figure this out without having to list everything.
I can figure this out for degree 3 in z3 just fine, but I'm having difficulties with degree 4.  Can someone please show me a step by step to possibly get an answer to this?
Thank you!
We have a test coming up, and I have a feeling this is going to be on it, and I'm trying to understand this completely.

I don’t see how to do it without any listing at all. I would do it this way, but I’d be using a fairly primitive symbolic-computation package to help me. First, I’d work over $k=\Bbb F_9$, and find an element $z\in S=\Bbb F_{81}\setminus\Bbb F_9$. Then I would do a listing of the elements of $S$, namely all $a+bz$ with $a,b\in k$ but $a\ne0$. Then I’d find all $4$-tuples $A_y=\{y,y^3,y^9,y^{27}\}$, taking care not to repeat any. Since I started with $72=81-9$ things in $S$, there’d be $72/4=18$ disjoint sets $A_y$. Then I’d multiply out $(X-y)(X-y^3)(X-y^9)(X-y^{27})\in\Bbb F_3[X]$ to get the irreducible quartic corresponding to each $A_y$.
