Define $F[x]/(q)$ for polynomial $q$ by $f=g$ $ mod(q) \iff q|(f-g) $. Denote the equivalence classes to be $[f]_q$ and the operations $[f]_q+[g]_q=[f+g]_q$ and $[f]_q*[g]_q=[f*g]_q$ (Sorry, I'm too new to know how much of this canonical notation).
Prove that the set of polynomials $F[x]/(q)$ is a field if and only if $q$ is irreducible.
I frankly have no idea how to proceed. I tried going back to the corresponding one in modular arithmetic, but no luck.