View source for Quadratically closed field

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A [[field]] in which every element of the field has a [[square root]] in the field.<ref name=Lam33>Lam (2005) p.&nbsp;33</ref><ref name=R230>Rajwade (1993) p.&nbsp;230</ref>

===Examples===
* The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]] is quadratically closed.
* The field of real numbers is not quadratically closed as it does not contain a square root of $-1$.
* The union of the [[finite field]]s $F_{5^{2^n}}$ for $n \ge 0$ is quadratically closed but not algebraically closed.<ref name=Lam34/>
* The field of [[constructible number]]s is quadratically closed but not algebraically closed.<ref name=Lam220>Lam (2005) p.&nbsp;220</ref>

===Properties===
* A field is quadratically closed if and only if it has [[universal invariant]] equal to 1.
* Every quadratically closed field is a [[Pythagorean field]] but not conversely (for example, $\mathbb{R}$ is Pythagorean); however, every non-[[Formally real field|formally real]] Pythagorean field is quadratically closed.<ref name=R230/>
* A field is quadratically closed if and only if its [[Witt–Grothendieck ring]] is isomorphic to $\mathbb{Z}$ under the dimension mapping.<ref name=Lam34>Lam (2005) p.&nbsp;34</ref>
* A formally real [[Euclidean field]] $E$ is not quadratically closed (as $-1$ is not a square in $E$) but the quadratic extension $E(\sqrt{-1})$ is quadratically closed.<ref name=Lam220/>
* Let $E/F$ be a finite [[field extension|extension]] where $E$ is quadratically closed.  Either $-1$ is a square in $F$ and $F$ is quadratically closed, or $-1$ is not a square in $F$ and $F$ is Euclidean.  This "going-down theorem" may be deduced from the [[Diller–Dress theorem]].<ref name=Lam270>Lam (2005) p.270</ref>

==Quadratic closure==
A '''quadratic closure''' of a field $F$ is a quadratically closed field which embeds in any other quadratically closed field containing $F$.  A quadratic closure for a given $F$ may be constructed as a subfield of the [[algebraic closure]] $F^{\mathrm{alg}}$ of $F$, as the union of all quadratic extensions of $F$ in $F^{\mathrm{alg}}$.<ref name=Lam220/>

===Examples===
* The quadratic closure of the field of real numbers is the field of complex numbers.<ref name=Lam220/>
* The quadratic closure of the finite field $\mathbb{F}_5$ is the union of the $\mathbb{F}_{5^{2^n}}$.<ref name=Lam220/>
* The quadratic closure of the field of rational numbers is the field of constructible numbers.

==References==
<references/>
* Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'',  Graduate Studies in Mathematics '''67''',  American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZBL|1068.11023}} {{MR|2104929 }}
* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}}{{ZBL|0785.11022}}