Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain

Question

A commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain.

I have done the part that if $F$ is a field then $F[x]$ is a PID using the division algorithm and contradicting the minimality of degree of a polynomial.

But I am facing difficulty to do the other part.

Answer 1

Suppose $k[X]$ is a PID. Prove that $(X)$ is a maximal ideal and then note $k\simeq k[X]/(X)$.