Partition matroid: Difference between revisions

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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.&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 <math>F_{5^{2^n}}</math> for ''n''&nbsp;≥&nbsp;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, '''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.&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''(√−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 07:07, 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 F52n 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 F52n.[4]
  • The quadratic closure of Q is the field of constructible numbers.

References

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  1. ↑ Lam (2005) p. 33
  2. ↑ 2.0 2.1 Rajwade (1993) p. 230
  3. ↑ 3.0 3.1 Lam (2005) p. 34
  4. ↑ 4.0 4.1 4.2 4.3 4.4 Lam (2005) p. 220
  5. ↑ Lam (2005) p.270
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