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. 1.

$K$ is quadratically closed, and

2. 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 QuadraticClosure 2013-03-22 15:42:43 2013-03-22 15:42:43 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 12F05 quadratically closed