Let $F$ be a field , I want to prove that every proper nontrivial prime ideal of $F[x]$ is maximal.
My definitions of prime/maximal ideals are as follows:
$N$ is a prime ideal of $R$ iff $ab \in N \implies a \in N$ or $b \in N$.
Two definitions for maximal:
$I$ is a maximal ideal of $R$ if there is no proper ideal $N$ of $R$ properly containing $I$.
or
$I$ is a maximal ideal of a commutative ring $R$ iff $R/I$ is a field.
There is another definition my professor did in class involving cosets and units but I can't recall it.
As far as I am aware there are non-commutative fields, which makes this problem a little tricky.
Best Answer
Proof:
Let $I$ be a nontrivial prime ideal for $F[x]$. Since $F$ is a field, that means $F$ is a Euclidean Domain which also implies $F$ is a PID. So $I$ is a principal ideal which is generated by $f$ for some $f \in F[x].$
$I$ is maximal if and only if $f$ is irreducible.
Think you can go from there?