# Quadratically closed field

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In [mathematics](/source/mathematics), a '''quadratically closed field''' is a [field](/source/field_(mathematics)) of [characteristic](/source/Characteristic_(algebra)) not equal to 2 in which every element has a [square root](/source/square_root).<ref name=Lam33>Lam (2005) p.&nbsp;33</ref><ref name=R230>Rajwade (1993) p.&nbsp;230</ref>
==Examples==
* The field of [complex number](/source/complex_number)s is quadratically closed; more generally, any [algebraically closed field](/source/algebraically_closed_field) is quadratically closed.
* The field of [real numbers](/source/real_numbers) is not quadratically closed as it does not contain a square root of −1.
* The union of the [finite field](/source/finite_field)s <math>\mathbb F_{5^{2^n}}</math> for ''n''&nbsp;≥&nbsp;0 is quadratically closed but not algebraically closed.<ref name=Lam34/>

==Properties==
* A field is quadratically closed if and only if it has [universal invariant](/source/universal_invariant) equal to 1.
* Every quadratically closed field is a [Pythagorean field](/source/Pythagorean_field) but not conversely (for example, '''R''' is Pythagorean); however, every non-[formally real](/source/formally_real) Pythagorean field is quadratically closed.<ref name=R230/>
* A field is quadratically closed if and only if its [Witt–Grothendieck ring](/source/Witt%E2%80%93Grothendieck_ring) is [isomorphic](/source/Ring_homomorphism) to '''Z''' under the dimension mapping.<ref name=Lam34>Lam (2005) p.&nbsp;34</ref>
* A formally real [Euclidean field](/source/Euclidean_ordered_field) ''E'' is not quadratically closed (as −1 is not a square in ''E'') but the quadratic extension ''E''({{radic|−1}}) is quadratically closed.<ref name=Lam220>Lam (2005) p.&nbsp;220</ref>
* Let ''E''/''F'' be a finite [extension](/source/field_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](/source/Diller%E2%80%93Dress_theorem).<ref name=Lam270>Lam (2005) p.270</ref>

==Quadratic closure==
A '''quadratic closure''' of a field ''F'' is a quadratically closed field containing ''F'' which [embeds](/source/Embedding) in any quadratically closed field containing ''F''.  A quadratic closure for any given ''F'' may be constructed as a subfield of the [algebraic closure](/source/algebraic_closure) ''F''<sup>alg</sup> of ''F'', as the union of all iterated quadratic extensions of ''F'' in ''F''<sup>alg</sup>.<ref name=Lam220/>

===Examples===
* The quadratic closure of '''R''' is '''C'''.<ref name=Lam220/>
* The quadratic closure of <math>\mathbb F_5</math> is the union of the <math>\mathbb F_{5^{2^n}}</math>.<ref name=Lam220/>
* The quadratic closure of '''Q''' is the field of complex [constructible numbers](/source/Constructible_number).

==References==
{{reflist}}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[Graduate Studies in Mathematics](/source/Graduate_Studies_in_Mathematics) | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}

Category:Field theory

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