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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.

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Do you know some ring theory? For a ring $R$ and an ideal $I$, $R/I$ is a field if and only if $I$ is a maximal ideal. So you could show that the maximal ideals in $F[x]$ are precisely those generated by irreducible polynomials. – Jeff Jan 17 at 18:30
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Are you familiar with ideals and quotient rings? If not, then mimic the proof that $\Bbb Z/q$ is a field $\iff q$ is prime, using the fact that $F[x],\,$ like $\Bbb Z,\,$ enjoys a Euclidean algorithm. – Math Gems Jan 17 at 18:36
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By "tried going back"... does that mean you know how to prove $\mathbb{Z} / (p)$ is a field when $p$ is a prime? What is the most important idea in that proof? – Hurkyl Jan 17 at 18:38

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