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 $ℂ$ and ${𝔽}_2$.

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 $\tilde{k}$ 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.

• $ℂ=\widetilde{ℝ}$.

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

• If $k={𝔽}_5$, 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 $K=\tilde{k}$.