In Cassels' article "Global Fields", he uses the term "ring of integers" and the notation $\mathcal{O}_K$, where $K$ is a field with a non-archimedean valuation, to denote the ring of elements $x \in K$ with $|x| \leq 1$. How justified is this notation?
In the case that $K = \mathbb{Q}$ and $| \cdot |$ is a $p$-adic valuation, $\mathcal{O}_K$ is the localization of $\mathbb{Z}$ at $p$, i.e. $(\mathbb{Z})_p$. In the case that $K$ is a number field with a $\mathfrak{p}$-adic valuation, this is similarly the localization at $\mathfrak{p}$ of its ring of integers, so the definition is reasonable here.
What about complete fields? If $K = \mathbb{Q}_p$, $\mathcal{O}_K = \mathbb{Z}_p$. Is it true that for $K$ a finite extension of $\mathbb{Q}_p$, $\mathcal{O}_K$ is the integral closure of $\mathbb{Z}_p$ in $K$? Is this integral closure – which I'll denote $\mathbb{Z}_K$ – already a local ring, or is localization necessary?
I think I can prove it if $\mathbb{Z}_K$ is a DVR – as $(\mathcal{O}_K)_\mathfrak{p}$ is in the number field case – but I don't know why (or if) this should be true.
Here's my attempt to extend the proof from the number field case: let $\mathbb{Z}_K$ be the integral closure of $\mathbb{Z}_p$ in $K$. Then, it's easy to see that it is contained in $\mathcal{O}_K$ by applying the ultrametric inequality to an integral equation for an element. Also, $K$ is the fraction field of $\mathbb{Z}_K$: for any integral domain $A$ with field of fractions $F$, the field of fractions of the integral closure of $A$ in some finite extension $K$ of $F$ is just $K$ (as shown here). If we knew that $\mathbb{Z}_K$ were a DVR with uniformizer $\pi$, we could write an element $x$ of $K$ as $\frac{\pi^m u}{\pi^n v}$ with $u, v$ units of $\mathbb{Z}_K \subseteq \mathcal{O}_K$, and thus $|u| = |v| = 1$. Since the valuation is non-trivial on $\mathbb{Z}_K$, we'd have $|\pi|< 1$, so if $x \in \mathcal{O}_K$, we'd have $m \geq n$ and thus $x = \pi^k w \in \mathbb{Z}_K$ where $k \geq 0$ and $w$ is a unit of $\mathbb{Z}_K$.
EDIT Is the integral closure $\mathbb{Z}_K$ of $\mathbb{Z}_p$ in a finite extension $K$ of $\mathbb{Q}_p$ a DVR? The answer below seems to show that the $\mathfrak{p}$-adic completion of the ring of integers of a number field is a DVR. But are these the same? I know that the finite extension $K$ of $\mathbb{Q}_p$ is the $\mathfrak{p}$-adic completion of some number field $K_0$. Is the same true for the rings of integers?
Best Answer
I think there is a flaw in your attempt : you are assuming that the topology (induced by the DVR structure) on $\mathbb{Z}_K$ is the same as the norm topology, which has to be proved.
I have asked the same question here. The proof can be found in : Neukirch, 'Algebraic number theory' (first step of the proof of Theorem 4.8 of Chapter II). I can give some detail if you don't have access to the book.