Are the p-adic integers the ring of integers of the field of p-adic numbers?

Question

This question was much simpler, but as I was typing it, it became a chain of questions.

My starting question was

Is $\mathbb{Z}_p$ (obtained by the inverse limit procedure with the directed system $\cdots \to \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$) the integral closure of $\mathbb{Z}$ in $\mathbb{Q}_p$?

My Intuition: Since $\mathbb{Z}_p$ is uncountable, and "the integral closure of $\mathbb{Z}$ in $\mathbb{R}$"(The Algebraic integers?) is countable, maybe the integral closure of $\mathbb{Z}$ in $\mathbb{Q_p}$ should be countable too?

I was hoping to look for a ring which served $\mathbb{R}$ the same purpose as $\mathbb{Z}$ serves $\mathbb{Q}$. That is being integrally closed. As a side note, I also realized that I do not know any ring whose field of fractions is $\mathbb{R}$. Is it because:

• such a thing does not exist.
• such a thing exists only by application of AC.
• I'm being slow today.

Thanks for the help!

Answer 1

The ring of integers of $\mathbb{Q}_p$ cannot be $\mathbb{Z}_p$ because of the countability argument that you mention. However, if $\mathbb{Q}_p$ is given the topology inherited from the $p$-adic norm $|\hspace{1mm}|_p$, then $\mathbb{Z}_p$ is the closure (in the topological sense) of $\mathbb{Z}$ in $\mathbb{Q}_p$. A proof of the latter statement can be found in Juergen Neukirch's "Algebraic Number Theory", Chapter $2$, Proposition $2.3$.

Answer 2

To reconcile $O_K$ for $K$ either a number field or a local field (finite extension of $\Bbb{Q}_p$ or $\Bbb{F}_p((t))$)

you can define $O_K$ as $$\{ a\in K, \forall \text{ discrete valuation } v , v(a)\ge 0\}$$