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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> | | In Milan this unique Sept, the errant indicate by means of Prada must have been a actual News.<br>This helmet is made of super strong ABS Thermoplastic Shell that offers you the protection and durability that you need. It also has D ring fasteners for a more secure enclosure and is manufactured to meet and even exceed DOT standards for motorcycle helmets.<br>http://www.bendtrapclub.com/cheap/ugg.asp?p=358 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=430 <br /> http://www.bendtrapclub.com/cheap/ugg.asp? When you loved this article and you would love to receive more info with regards to [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] assure visit our site. p=46 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=223 <br /> http://www.bendtrapclub.com/cheap/ugg.asp?p=403 <br /> <br>http://atlantaka.org/?document_srl=926201<br>http://www.tinterovirtual.com/modules.php?name=Your_Account&op=userinfo&username=LWhitacre |
| ==Examples==
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| * The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]] is quadratically closed.
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| * The field of real numbers is not quadratically closed as it does not contain a square root of −1.
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| * 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/>
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| * The field of [[constructible number]]s is quadratically closed but not algebraically closed.<ref name=Lam220>Lam (2005) p. 220</ref>
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| ==Properties==
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| * A field is quadratically closed if and only if it has [[universal invariant]] equal to 1.
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| * 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/>
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| * 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>
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| * 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/>
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| * 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>
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| ==Quadratic closure==
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| 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/>
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| ===Examples===
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| * The quadratic closure of '''R''' is '''C'''.<ref name=Lam220/>
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| * The quadratic closure of '''F'''<sub>5</sub> is the union of the <math>F_{5^{2^n}}</math>.<ref name=Lam220/>
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| * The quadratic closure of '''Q''' is the field of constructible numbers.
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| ==References==
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| {{reflist}}
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| * {{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 }}
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| * {{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 }}
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| [[Category:Field theory]]
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