---
tag: H20P600HQS
kind: concept
name: Integrally closed domain
msc: [13B22, 13G05]
specializes: [0QVD7WBFGN]
sources:
  - url: https://en.wikipedia.org/wiki/Integrally_closed_domain
same_as:
  wikidata: Q2851427
  wikipedia: Integrally closed domain
---

## Summary

An integrally closed domain, or normal domain, is an [[integral domain|0QVD7WBFGN]] that contains every element of its [[field of fractions|0F2PTGX1A2]] that is [[integral|1EHN4KPSFS]] over it.
Every [[unique factorisation domain|0W4X9G6263]] is integrally closed, and so is every Dedekind domain and every valuation ring.
The [[going-down theorem|YE52MQ13MV]] holds over integrally closed domains.
In geometry, integrally closed domains are the coordinate rings of normal varieties, whose singular points form a set of codimension at least two.

## Explanations

### L1 human

A fraction that satisfies a monic equation with coefficients in the ring should already be in the ring.
For the integers this is the familiar fact that a rational root of $x^n + a_{n-1} x^{n-1} + \cdots + a_0$ with integer coefficients is an integer.
The ring $k[t^2, t^3]$ of polynomials without a linear term fails this test.
The fraction $t = t^3 / t^2$ is a root of $x^2 - t^2$, a monic polynomial over the ring, but $t$ itself is not in the ring.
Geometrically this ring describes the curve $y^2 = x^3$, which has a cusp at the origin.