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 $$k\le k(\sqrt{2})\le k(\sqrt[4]{2})\le \cdots \le k(\sqrt[2^n]{2})\le \cdots $$ Take the union of all these fields to obtain a field $K$ Then it can be shown that $K=\widetilde{k}$