$\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory

Question

I want to show that $\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory.

I know that if the class group is trivial, then it is a PID. Is this helpful to show the claim or how else can I proceed?

I'd be thankful for any help!