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{{DISPLAYTITLE:''u''-invariant}}
In mathematics, the '''universal invariant''' or '''''u'''''<nowiki></nowiki>'''-invariant''' of a [[field (mathematics)|field]] describes the structure of [[quadratic form]]s over the field. 
 
The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an [[anisotropic quadratic space]] over ''F'', or ∞ if this does not exist.  Since [[formally real field]]s have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields.  An equivalent formulation is that ''u'' is the smallest number such that every form of dimension greater than ''u'' is [[Isotropic quadratic form|isotropic]], or that every form of dimension at least ''u'' is [[Universal quadratic form|universal]].
 
==Examples==
* For the complex numbers, ''u''('''C''') = 1.
* If ''F'' is [[Quadratically closed field|quadratically closed]] then ''u''(''F'') = 1.
* The function field of an [[algebraic curve]] over an [[algebraically closed field]] has ''u'' ≤ 2; this follows from [[Tsen's theorem]] that such a field is [[quasi-algebraically closed]].<ref name=Lam376>Lam (2005) p.376</ref>
* If ''F'' is a nonreal [[global field|global]] or [[local field]], or more generally a [[linked field]], then ''u''(''F'') = 1,2,4 or 8.<ref name=Lam406>Lam (2005) p.406</ref>
 
==Properties==
* If ''F'' is not formally real then ''u''(''F'') is at most <math>q(F) = \left|{F^\star / F^{\star2}}\right|</math>, the index of the squares in the multiplicative group of ''F''.<ref name=Lam400>Lam (2005) p.&nbsp;400</ref>
* Every even integer occurs as the value of ''u''(''F'') for some ''F''.<ref name=Lam402>Lam (2005) p.&nbsp;402</ref>
* ''u''(''F'') cannot take the values 3, 5, or 7.<ref name=Lam401>Lam (2005) p.&nbsp;401</ref>  A field exists with ''u''&nbsp;=&nbsp;9.<ref>{{cite journal | title=Fields of u-Invariant 9 | first=Oleg T. | last=Izhboldin | journal=[[Annals of Mathematics]], 2 ser | volume=154 | number=3 | year=2001 | pages=529–587 | url=http://www.jstor.org/stable/3062141 | zbl=0998.11015 }}</ref>
 
==The general ''u''-invariant==
Since the ''u''-invariant is of little interest in the case of formally real fields, we define a '''general''' '''''u'''''<nowiki></nowiki>'''-invariant''' to be the maximum dimension of an anisotropic form in the [[torsion subgroup]] of the [[Witt ring (forms)|Witt ring]] of '''F''', or ∞ if this does exist.<ref name=Lam409>Lam (2005) p.&nbsp;409</ref>  For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition.<ref name=Lam410>Lam (2005) p.&nbsp;410</ref>  For a formally real field, the general ''u''-invariant is either even or ∞.
 
===Properties===
* ''u''(''F'') ≤ 1 if and only if ''F'' is a [[Pythagorean field]].<ref name=Lam410/>
 
==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 }}
 
{{algebra-stub}}
 
[[Category:Field theory]]
[[Category:Quadratic forms]]

Revision as of 18:07, 16 August 2013

In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field.

The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that u is the smallest number such that every form of dimension greater than u is isotropic, or that every form of dimension at least u is universal.

Examples

Properties

  • If F is not formally real then u(F) is at most q(F)=|F⋆/F⋆2|, the index of the squares in the multiplicative group of F.[3]
  • Every even integer occurs as the value of u(F) for some F.[4]
  • u(F) cannot take the values 3, 5, or 7.[5] A field exists with u = 9.[6]

The general u-invariant

Since the u-invariant is of little interest in the case of formally real fields, we define a general u-invariant to be the maximum dimension of an anisotropic form in the torsion subgroup of the Witt ring of F, or ∞ if this does exist.[7] For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition.[8] For a formally real field, the general u-invariant is either even or ∞.

Properties

References

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Template:Algebra-stub

  1. ↑ Lam (2005) p.376
  2. ↑ Lam (2005) p.406
  3. ↑ Lam (2005) p. 400
  4. ↑ Lam (2005) p. 402
  5. ↑ Lam (2005) p. 401
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  7. ↑ Lam (2005) p. 409
  8. ↑ 8.0 8.1 Lam (2005) p. 410
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