Relationship Between Ring of Integers of a Number Field to P-adic integers

Question

Suppose we have a number field $K$ with ring of integers $\mathcal O_K$. Let $\frak p$ be a prime ideal in $K$ lying over $p\in \mathbb Q$. Then, using the $\frak p$-adic norm, we may define the $p$-adic ring of integers $\mathbb Z_{\frak p} = \{ x\in K: |x|_{\frak p}\leq 1\}$. I know that $\mathcal O_K \not= \mathbb Z_{\frak p}$, but is $\mathcal O_K \subset \mathbb Z_{\frak p}$?

Answer 1

Yes. As a DVR $\Bbb{Z}_{\mathfrak{p}}$ is integrally closed in its field of fractions $K$. It contains $\Bbb{Z}$ so it also contains the integral closure of $\Bbb{Z}$ in $K$ which is ...