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Ring of integers of a number field
For a number field $K_{/ {{\mathbf{Q}}}} $, define the ring of integers as \begin{align*}{\mathcal{O}}_K \coloneqq \operatorname{cl}^{\mathrm{int}} _K({\mathbf{Z}}).\end{align*} More generally, for \(R\) a ring, \begin{align*} {\mathcal{O}}_R \coloneqq \operatorname{cl}^{\mathrm{int}} _{\operatorname{ff}(R)}(R) \end{align*} or equivalently a maximal order.
Ring of integers of a nonarchimedean field
For a nonarchimedean field with an absolute value \({\left\lvert {{-}} \right\rvert}\), say induced by a valuation, the ring of integers is given by the valuation ring, i.e. the closed disc: \begin{align*} {\mathcal{O}}_K \coloneqq\left\{{ x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} \leq 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) \geq 0}\right\} = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \subseteq K \end{align*} Its units are given by the boundary sphere: \begin{align*} {\mathcal{O}}_K^{\times}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} = 1}\right\} = \left\{{x\in K {~\mathrel{\Big\vert}~}v(x) = 0}\right\} = {{\partial}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu_K \end{align*} This is a local ring with maximal ideal the open disc: \begin{align*} {\mathfrak{m}}\coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert} < 1 }\right\}= {\mathbb{D}}_K \end{align*}
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