nt.number theory - Quick proof of the fact that the ring of integers of Q(zeta_n) is Z[zeta_n]?

The ring of integers $R_n$ of ${mathbf Q}(zeta_n)$ contains ${mathbf Z}[zeta_n]$ as a subring with finite index. To show the containment of rings is an equality, it suffices to show the inclusion ${mathbf Z}[zeta_n] rightarrow R_n$ becomes an isomorphism after tensoring with ${mathbf Z}_p$ for all $p$, and this basically boils down to showing the ring of integers of ${mathbf Q}_p(zeta_n)$ is ${mathbf Z}_p[zeta_n]$ for all $p$. Now you have to know something about how to compute rings of integers in unramified and totally ramified extensions of local fields. I'm leaving off some details here, admittedly, and since I used the terrible word "compute" maybe this isn't an answer you are looking for.

You didn't tell us whether you were okay with the inductive argument but disliked (apparently) the prime-power case or you were unhappy with both aspects.