View source for Universal invariant of a field

{{TEX|done}}{{MSC|15A63}}

''$u$-invariant''

A numerical  invariant of a [[field]] describing 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 form|anisotropic quadratic space]] over $F$, or $\infty$ 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 $u(F)$ 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 a [[universal quadratic form]].

===Examples===
* For the field of [[complex number]]s, $u(\mathbb{C}) = 1$.
* A [[quadratically closed field]] has $u = 1$.
* The function field of an [[algebraic curve]] over an [[algebraically closed field]] has $u \le 2$; this follows from [[Tsen's theorem]] that such a field is [[Quasi-algebraically closed field|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 $u(F) = 1,2,4 \,\text{or}\, 8$.<ref name=Lam406>Lam (2005) p.406</ref>

===Properties===
* If $F$ is not formally real then $u(F)$ is at most $q(F) = \left|{F^\star / F^{\star2}}\right|$, the index of the subgroup of 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>Izhboldin (2001)</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]] of the [[Witt ring (forms)|Witt ring]] of $F$, or $\infty$ 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 $\infty$.

===Properties===
* $u(F) \le 1$ if and only if $F$ is a [[Pythagorean field]].<ref name=Lam410/>

==References==
<references/>
* Oleg T. Izhboldin, "Fields of u-Invariant 9", ''Annals of Mathematics'', 2 ser  '''154''':3 (2001) pp.529–587 [http://www.jstor.org/stable/3062141] {{ZBL|0998.11015}}
* Tsit Yuen Lam, ''Introduction to Quadratic Forms over Fields'',  Graduate Studies in Mathematics '''67''',  American Mathematical Society (2005) {{ISBN|0-8218-1095-2}} {{ZBL|1068.11023}} {{MR|2104929 }}
*  Albrecht Pfister, ''Quadratic forms with applications to algebraic geometry and topology'', London Mathematical Society Lecture Note Series '''217''' Cambridge University Press (1995) {{ISBN|0-521-46755-1}} {{ZBL|0847.11014}}
* A. R. Rajwade, ''Squares'', London Mathematical Society Lecture Note Series '''171''' Cambridge University Press (1993) {{ISBN|0-521-42668-5}} {{ZBL|0785.11022}}