Prove that $I$ is a maximal ideal

Question

I have a question. To show that the ideal $I=\langle f(x)\rangle $ is a maximal ideal of $K[x]$ do I have to show that $f(x)$ is irreducible in $K[x]$? Or is there an other way to prove that $I$ is a maximal ideal?

Answer 1

Yes, if $K$ is a field $K[x]$ is a PID, and there the prime elements and irreducible elements coincide, and they are the generators of maximal ideals.

Answer 2

This is a fairly broad question - there could be lots of ways of proving something depending on the situation. As rschwieb points out, proving that $I$ is maximal is equivalent to proving $f$ is irreducible, but that doesn't mean you have to prove $f$ is irreducible first. You could prove that $K[x]/I$ is a field, which would imply $I$ is maximal (and therefore that $f$ is irreducible).