There's this exercise that really has kept me stuck for a day by now, will you please help me figure out:
let's consider polynomials in $\mathbb Z_3$:
- characterize degree 2 not irreducible monic polynomials. How many are they?
- characterize degree 2 irreducible monic polynomials. How many are they?
- how many degree 4 monic polynomials have no roots and are not irreducible?
I have no idea how to characterize those polynomials, I just figured that all polynomials with $\Delta = 0$ or $\Delta = 1$ are not irreducible, if $\Delta = 2$ then the polynomial is actually irreducible. I couldn't help but enumerate all monic degree 2 polynomials and find one by one for myself if they met the condition I've set. Is there a better way to count them without being forced to find them all? What about the last question, that really left me with no idea.