---
date: 2022-01-24 18:47
modification date: Monday 24th January 2022 18:47:45
title: ring of integers
aliases: [ring of integers]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #NT/algebraic 
- Refs:
	- #todo/add-references
- Links:
	- [valuation](Unsorted/Valuations.md)
	- [absolute value](Unsorted/absolute%20value.md)
	- [archimedean](Unsorted/absolute%20value.md)
	- [number field](Unsorted/number%20field.md)

---


# Ring of integers of a number field

For a [number field](number%20field.md) $K\slice \QQ$, define the **ring of integers** as  $$\OO_K \da \intcl_K(\ZZ).$$
More generally, for $R$ a ring,
$$
\OO_R \da \intcl_{\ff(R)}(R)
$$
or equivalently a [maximal order](Unsorted/order.md).

# Ring of integers of a nonarchimedean field

For a [nonarchimedean field](Unsorted/absolute%20value.md) with an [absolute value](Unsorted/absolute%20value.md) $\abs{\wait}$, say induced by a [valuation](Unsorted/Valuations.md), the **ring of integers** is given by the [valuation ring](valuation%20ring.md), i.e. the closed disc:
$$
\OO_K \da \ts{ x\in K \st \abs{x} \leq 1} = \ts{x\in K \st v(x) \geq 0} = \bar{\DD}_K \subseteq K 
$$
Its units are given by the boundary sphere:
$$
\OO_K\units \da \ts{x\in K \st \abs{x} = 1} = \ts{x\in K \st v(x) = 0} =  \bd\bar{\DD}_K
$$
This is a [local ring](Unsorted/localization%20of%20rings.md) with maximal ideal the open disc:
$$
\mfm \da \ts{x\in K \st \abs{x} < 1 }= \DD_K 
$$

# Misc
![](attachments/Pasted%20image%2020220224124633.png)