# U-invariant

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/U-invariant
> Markdown URL: https://mediated.wiki/source/U-invariant.md
> Source: https://en.wikipedia.org/wiki/U-invariant
> Source revision: 1345533138
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{DISPLAYTITLE:''u''-invariant}}
In [mathematics](/source/mathematics), the '''universal invariant''' or '''''u''-invariant''' of a [field](/source/field_(mathematics)) describes the structure of [quadratic form](/source/quadratic_form)s over the field.

The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an [anisotropic quadratic space](/source/anisotropic_quadratic_space) over ''F'', or ∞ if this does not exist.  Since [formally real field](/source/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](/source/Isotropic_quadratic_form), or that every form of dimension at least ''u'' is [universal](/source/Universal_quadratic_form).

==Examples==
* For the [complex number](/source/complex_number)s, ''u''('''C''') = 1.
* If ''F'' is [quadratically closed](/source/Quadratically_closed_field) then ''u''(''F'') = 1.
* The function field of an [algebraic curve](/source/algebraic_curve) over an [algebraically closed field](/source/algebraically_closed_field) has ''u'' ≤ 2; this follows from [Tsen's theorem](/source/Tsen's_theorem) that such a field is [quasi-algebraically closed](/source/quasi-algebraically_closed).<ref name=Lam376>Lam (2005) p.376</ref>
* If ''F'' is a non-real [global](/source/global_field) or [local field](/source/local_field), or more generally a [linked field](/source/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 and the characteristic of ''F'' is not ''2'' 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](/source/group_(mathematics)) of ''F''.<ref name=Lam400>Lam (2005) p.&nbsp;400</ref>
* ''u''(''F'') cannot take the values 3, 5, or 7.<ref name=Lam401>Lam (2005) p.&nbsp;401</ref>  Fields exist with ''u''&nbsp;=&nbsp;6<ref name=Lam484>Lam (2005) p.484</ref><ref name=Lam1989>{{cite book | last=Lam | first=T.Y. | authorlink=Tsit Yuen Lam | chapter=Fields of u-invariant 6 after A. Merkurjev | zbl=0683.10018 | title=Ring theory 1989. In honor of S. A. Amitsur, Proc. Symp. and Workshop, Jerusalem 1988/89 | series=Israel Math. Conf. Proc. | volume=1 | pages=12–30 | year=1989 }}</ref> and ''u''&nbsp;=&nbsp;9.<ref>{{cite journal | title=Fields of u-Invariant 9 | first=Oleg T. | last=Izhboldin |authorlink1=Oleg Izhboldin| journal=Annals of Mathematics |series=Second Series | volume=154 | number=3 | year=2001 | pages=529–587 | doi=10.2307/3062141 | jstor=3062141 | zbl=0998.11015 }}</ref>
* [Merkurjev](/source/Alexander_Merkurjev) has shown that every [even](/source/parity_(mathematics)) integer occurs as the value of ''u''(''F'') for some ''F''.<ref name=Lam402>Lam (2005) p.&nbsp;402</ref><ref name=ELM170>Elman, Karpenko, Merkurjev (2008) p.&nbsp;170</ref>
* [Alexander Vishik](/source/Alexander_Vishik) proved that there are fields with ''u''-invariant <math>2^r+1</math> for all <math>r > 3</math>.<ref>{{cite book | last=Vishik | first=Alexander  | title=Algebra, Arithmetic, and Geometry | chapter=Fields of ''u''-invariant <math>2^r + 1</math> | series=Progress in Mathematics | year=2009 | volume=270 | pages=661–685 | doi = 10.1007/978-0-8176-4747-6_22 | publisher=Birkhäuser Boston | isbn=978-0-8176-4746-9 }}</ref>
* The ''u''-invariant is bounded under finite-[degree](/source/Degree_of_a_field_extension) [field extension](/source/field_extension)s.  If ''E''/''F'' is a field extension of degree ''n'' then
::<math>u(E) \le \frac{n+1}{2} u(F) \ . </math>
In the case of quadratic extensions, the ''u''-invariant is bounded by
:<math>u(F) - 2 \le u(E) \le \frac{3}{2} u(F) \ </math>
and all values in this range are achieved.<ref>{{cite book | last1=Mináč | first1=Ján | last2=Wadsworth | first2=Adrian R. | chapter=The u-invariant for algebraic extensions | zbl=0824.11018 | editor1-first=Alex  | editor1-last=Rosenberg|editor1-link=Alex F. T. W. Rosenberg | title=K-theory and algebraic geometry: connections with quadratic forms and division algebras. Summer Research Institute on quadratic forms and division algebras, July 6-24, 1992, University of California, Santa Barbara, CA (USA) | location=Providence, RI | publisher=[American Mathematical Society](/source/American_Mathematical_Society) | series=Proc. Symp. Pure Math. | volume=58 | pages=333–358 | year=1995 | issue=2 }}</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''-invariant''' to be the maximum dimension of an anisotropic form in the [torsion subgroup](/source/torsion_subgroup) of the [Witt ring](/source/Witt_ring_(forms)) of '''F''', or ∞ if this does not 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'') ≤&thinsp;1 if and only if ''F'' is a [Pythagorean field](/source/Pythagorean_field).<ref name=Lam410/>

==References==
{{reflist}}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[Graduate Studies in Mathematics](/source/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](/source/Cambridge_University_Press) | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
* {{cite book | title=The algebraic and geometric theory of quadratic forms | volume=56 | series=American Mathematical Society Colloquium Publications | first1=Richard | last1=Elman | first2=Nikita | last2=Karpenko | first3=Alexander | last3=Merkurjev | publisher=American Mathematical Society, Providence, RI | year=2008 | isbn=978-0-8218-4329-1 }}

Category:Field theory
Category:Quadratic forms

---
Adapted from the Wikipedia article [U-invariant](https://en.wikipedia.org/wiki/U-invariant) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/U-invariant?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.