Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal.
Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any suggestions on this would be appreciated.
[Math] Non-zero prime ideals of $F[x]$ are maximal
abstract-algebraidealsring-theory
Best Answer
Yes.
Hint: Every prime ideal $P$ of $F[x]$ is of the form $P=(f(x))$ for some polynomial $f$. Use this to show that $F[X]/P$ is a domain which is a finite dimensional vector space, and so a field (why?).