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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.

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

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