# Reference request - adic completions of number fields

For $$p$$-adic integers we have the following: The only maximal ideal of $$\mathbb{Z}_p$$ is $$p\mathbb{Z}_p$$ and $$\mathbb{Z}_p/p\mathbb{Z}_p\cong \mathbb{Z}/p\mathbb{Z}$$.

Suppose now we have a number field $$K$$ and a prime ideal $$\mathfrak{p}$$ of the ring of integers $$\mathcal{O}_K$$. Let $$|\cdot|_\mathfrak{p}$$ be the valuation/metric induced on $$K$$ and let $$K_\mathfrak{p}$$ denote the completion of $$K$$ w.r.t. this metric. Let $$\mathcal{O}_\mathfrak{p}=\{x\in K_\mathfrak{p}: |\cdot|_\mathfrak{p}\leq1\}$$. Then $$\mathfrak{p}\mathcal{O}_\mathfrak{p}=\{x\in K_\mathfrak{p}: |\cdot|_\mathfrak{p}<1\}$$ is a maximal ideal of $$\mathcal{O}_\mathfrak{p}$$ (right?); is it true that $$\mathcal{O}_\mathfrak{p}/\mathfrak{p}\mathcal{O}_\mathfrak{p}\cong \mathcal{O}_K/\mathfrak{p}$$? If so, what is a good reference?

I would appreciate any suggestions on a good introduction to adic completions of number fields.

• This is proved as Proposition II.4.3 in Neukirch’s Algebraic Number Theory. – Paul LeVan Oct 20 '18 at 3:44