quadratic closure
A field $K$ is said to be quadratically closed if it has no quadratic extensions. In other words, every element of $K$ is a square. Two obvious examples are $\u2102$ and ${\mathbb{F}}_{2}$.
A field $K$ is said to be a quadratic closure of another field $k$ if

1.
$K$ is quadratically closed, and

2.
among all quadratically closed subfields^{} of the algebraic closure^{} $\overline{k}$ of $k$, $K$ is the smallest one.
By the second condition, a quadratic closure of a field is unique up to field isomorphism. So we say the quadratic closure of a field $k$, and we denote it by $\stackrel{~}{k}$ Alternatively, the second condition on $K$ can be replaced by the following:
$K$ is the smallest field extension over $k$ such that, if $L$ is any field extension over $k$ obtained by a finite number of quadratic extensions starting with $k$, then $L$ is a subfield of $K$.
Examples.

•
$\u2102=\stackrel{~}{\mathbb{R}}$.

•
If $\mathbb{E}$ is the Euclidean field in $\mathbb{R}$, then the quadratic extension $\mathbb{E}(\sqrt{1})$ is the quadratic closure $\stackrel{~}{\mathbb{Q}}$ of the rational numbers^{} $\mathbb{Q}$.

•
If $k={\mathbb{F}}_{5}$, consider the chain of fields
$$k\le k(\sqrt{2})\le k(\sqrt[4]{2})\le \mathrm{\cdots}\le k(\sqrt[{2}^{n}]{2})\le \mathrm{\cdots}$$ Take the union of all these fields to obtain a field $K$. Then it can be shown that $K=\stackrel{~}{k}$.
Title  quadratic closure 

Canonical name  QuadraticClosure 
Date of creation  20130322 15:42:43 
Last modified on  20130322 15:42:43 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 12F05 
Defines  quadratically closed 