Let $p$ be a prime and $\xi\_n$ a nth primitive root of unity. What is the ring of integers of $\mathbb{Q}\_p(\xi\_n)$? Is it $\mathbb{Z}\_p[\xi\_n]$?



ring of integers of a cyclotomic extension of $\mathbb{Q}_p$
Yes. This follows e.g. from Propositions 16 and 17 in Section IV.4 of Serre: Local fields.


Indeed, let $n=p^mn'$, where $(n',p)=1$. By Proposition 17, $\mathbb{Q}\_p(\zeta\_{p^m})$ has ring of integers $\mathbb{Z}\_p[\zeta\_{p^m}]$. Then, applying Proposition 16 to the local field $\mathbb{Q}\_p(\zeta\_{p^m})$, we see that $\mathbb{Q}\_p(\zeta\_{p^m},\zeta\_{n'})$ has ring of integers $\mathbb{Z}\_p[\zeta\_{p^m},\zeta\_{n'}]$. However, it is straightforward that $\mathbb{Q}\_p(\zeta\_{p^m},\zeta\_{n'})=\mathbb{Q}\_p(\zeta\_{n})$ and $\mathbb{Z}\_p[\zeta\_{p^m},\zeta\_{n'}]=\mathbb{Z}\_p[\zeta\_{n}]$, hence $\mathbb{Q}\_p(\zeta\_{n})$ has ring of integers $\mathbb{Z}\_p[\zeta\_{n}]$.


