---
date: 2022-01-23 18:58
modification date: Sunday 23rd January 2022 18:58:28
title: integrally closed
aliases: ["integrally closed", "Integrally closed", "integral closure", "integral", "ring of integers", "order", "algebraic integers", "integral extension"]

---

Tags: 
#NT/algebraic #CA 
Refs:
[normalization](Unsorted/normalization.md)
[Noether normalization](Unsorted/Noether%20normalization.md)

# Integral extensions

![](attachments/Pasted%20image%2020220123190624.png)


![](attachments/Pasted%20image%2020220120134951.png)
# Integrally closed

![](attachments/Pasted%20image%2020220123185848.png)
![](attachments/Pasted%20image%2020220123190009.png)

![](attachments/Pasted%20image%2020220123190223.png)
![](attachments/Pasted%20image%2020220123190237.png)

# Integral Closure

#todo 

# Examples

Write $\intcl_B(A)$ for the integral closure of $A$ in $B$.
 
- $\intcl\QQ(\ZZ) = \ZZ$
- $\intcl_K(\ZZ) \da \OO_K$ for $K\in\Numberfield$.
- $\intcl_{\QQ[i]}(\ZZ) = \OO_{\QQ[i]} = \ZZ[m]$ where $m= {1\over 2}\qty{1 + \sqrt 5}$.
- $\intcl_{\QQ[\zeta_n]} = \OO_{\QQ[\zeta_n]} = \ZZ[\zeta_n]$
- $\intcl_\CC(\ZZ) = \algcl(\QQ)$, the algebraic closure of $\QQ$ or ring of **algebraic integers**.
 
![](attachments/Pasted%20image%2020220124092703.png)


# Exercises

- Show that integral extensions satisfy the [Cohen–Seidenberg theorems](Cohen–Seidenberg%20theorems).
- Show that if $B$ is integral over $A$, then $B \otimes_{A} R$ is integral over $\mathrm{R}$ for any A-algebra R.
- Show that if _A_ is a subring of a field _K_, then the integral closure of _A_ in _K_ is the intersection of all [valuation rings](Unsorted/Valuations.md) of _K_ containing _A_.
- Show that every UFD is integrally closed.
- For $A \leq B$ a subring, show that $b\in B$ is integral over $A$ iff there exists a faithful $A[b]\dash$submodule of $B$ which is finitely generated as an $A\dash$module.
	Hint: use Cramer's rule.
- Show that an $A\dash$algebra $B$ is finite iff it is finitely generated as an $A\dash$algebra by a generating set that is integral over $A$.
- Show that an $A\dash$algebra $B$ is finite iff finitely generated and integral over $A$.
- Show that UFDs are integrally closed.
-  Let $A$ be a normal integral domain, and let $E$ be a finite extension of the field of fractions $F$ of $A$. 
	Show that element of $E$ is integral over $A$ iff its minimum polynomial over $F$ has coefficients in $A$.
- What are the [integral](integral) elements of $\QQ\slice \ZZ$?
- Show that if $A\leq B$ is a submodule and $x_i\in B$ are integral over $A$, then $A[x_1,\cdots, x_n]$ is a finitely generated $A\dash$module.
- Show that [finite type](finite%20type.md) and integral implies finite.
- Show that integraility is transitive. Show that this also holds for being integrally closed.
- Show that integrality is preserved by passing to quotients or the ring of fractions.
- Show that integral closure is local.
- Show that if $\mfp\in \spec \kxn$  and $A \da \kxn/\mfp$ is a 1-dimensional domain, then $A$ is an integral extension of $k[x]$.
- Suppose $R$ is a [Noetherian](Noetherian.md) integral domain. Show that $R$ is a UFD iff $A$ is [integrally closed](integrally%20closed.md) in $\ff(R)$ and $\Cl(X) = 0$.
- Show that for $A\subseteq k$ a subring of a field, the integral closure $\cl_k(A)$ is the smallest valuation ring in $k$ containing $A$.
- Show that if $B$ is integrally closed ensures that every prime $\mathfrak{p}$ of $\mathcal{O}$ has at least one prime $\mathfrak{q}$ lying above it (this is a standard fact of commutative algebra).