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In mathematics, a '''quadratically closed field''' is a [[field (mathematics)|field]] in which every element of the field has a [[square root]] in the field.<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 field]]s <math>F_{5^{2^n}}</math> for ''n'' ≥ 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. 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, '''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''(√−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 any given ''F'' may be constructed as a subfield of the [[algebraic closure]] ''F''<sup>alg</sup> of ''F'', as the union of all 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 '''F'''<sub>5</sub> is the union of the <math>F_{5^{2^n}}</math>.<ref name=Lam220/> | |||
* The quadratic closure of '''Q''' is the field of 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]] | |||
Revision as of 08:07, 26 September 2012
In mathematics, a quadratically closed field is a field in which every element of the field has a square root in the field.[1][2]
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 for n ≥ 0 is quadratically closed but not algebraically closed.[3]
- The field of constructible numbers is quadratically closed but not algebraically closed.[4]
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.[2]
- A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to Z under the dimension mapping.[3]
- A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(√−1) is quadratically closed.[4]
- 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.[5]
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 any given F may be constructed as a subfield of the algebraic closure Falg of F, as the union of all quadratic extensions of F in Falg.[4]
Examples
- The quadratic closure of R is C.[4]
- The quadratic closure of F5 is the union of the .[4]
- The quadratic closure of Q is the field of constructible numbers.
References
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