Cyclotomic integers

Question

Let $K = \mathbb{Q}(\zeta_{p^{\infty}})$ be the field obtained by adjoining all $p$th power roots of unity to $\mathbb{Q}$. What is the ring of integer of $K$? Since the ring of integer of $\mathbb{Q}(\zeta_{p^n})$ is just $\mathbb{Z}[\zeta_{p^n}]$, should $\mathcal{O}_K$ just be $\mathbb{Z}[\zeta_{p^{\infty}}]$?

Answer 1

It is clear that $\mathcal{O}_K$ contains $\mathbb{Z}[\zeta_{p^\infty}]$, so one only has to prove the other inclusion. Suppose that $\alpha \in K$ is integral over $\mathbb{Z}$, i.e. $\alpha \in \overline{\mathbb{Z}}$. By construction of $K$, $\alpha \in K$ implies that $\alpha \in \mathbb{Q}(\zeta_{p^n})$ for some $n$. But then we conclude that $$ \alpha \in \mathbb{Q}(\zeta_{p^n}) \cap \overline{\mathbb{Z}} = \mathbb{Z}[\zeta_{p^n}]. $$