I have a question regarding polynomials over $\mathbb F_3$ ($\mathbb Z/3\mathbb Z$). In a homework problem we have to check how many third degree polynomials there are over $\mathbb F_3$.
The question is whether there are exactly $54$ or not. I'm confused as to how to go about this.
I can see how there should be a finite number of possibilities, but I don't know how to prove that there is a certain number of them.
$$a x^3+b x^2+c x+d$$ Modulus $3$ coefficient $a$ can be $1$ or $2$, while $b,c,d$ have a range $(0..3)$
Thus they are $2\times 3\times 3\times 3=54$