In mathematics, a '''quadratically closed field''' is a field of characteristic not equal to 2 in which every element has a square root.<ref name=Lam33>Lam (2005) p. 33</ref><ref name=R230>Rajwade (1993) p. 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 fields <math>\mathbb F_{5^{2^n}}</math> for ''n'' ≥ 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 equal to 1. * Every quadratically closed field is a Pythagorean field but not conversely (for example, '''R''' is Pythagorean); however, every non-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 '''Z''' under the dimension mapping.<ref name=Lam34>Lam (2005) p. 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''({{radic|−1}}) is quadratically closed.<ref name=Lam220>Lam (2005) p. 220</ref> * Let ''E''/''F'' be a finite 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 containing ''F'' which embeds in any quadratically closed field containing ''F''. A quadratic closure for any given ''F'' may be constructed as a subfield of the 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.
==References== {{reflist}} * {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=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 | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
Category:Field theory