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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 $\mathbb{Q}$.
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