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 $\mathbb{C}$ 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 $\widetilde{k}$ 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.

•

$\mathbb{C}=\widetilde{\mathbb{R}}$ .
•

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

If $k=\mathbb{F}_{5}$ , consider the chain of fields

$k\leq k(\sqrt{2})\leq k(\sqrt[4]{2})\leq\cdots\leq k(\sqrt[2^{n}]{2})\leq\cdots$

Take the union of all these fields to obtain a field $K$ . Then it can be shown that $K=\widetilde{k}$ .

Title	quadratic closure
Canonical name	QuadraticClosure
Date of creation	2013-03-22 15:42:43
Last modified on	2013-03-22 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