Universal invariant of a field

2020 Mathematics Subject Classification: Primary: 15A63 [MSN][ZBL]

$u$-invariant

A numerical invariant of a field describing 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 $\infty$ 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 $u(F)$ 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 a universal quadratic form.

Examples

• For the field of complex numbers, $u(\mathbb{C}) = 1$.
• 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.
• If $F$ is a nonreal global or local field, or more generally a linked field, then $u(F) = 1,2,4 \,\text{or}\, 8$.

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 $F$.
• Every even integer occurs as the value of $u(F)$ for some $F$.
• $u(F)$ cannot take the values 3, 5, or 7. A field exists with u = 9.

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 $\infty$ if this does not exist. For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition. 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.

References

• Oleg T. Izhboldin, "Fields of u-Invariant 9", Annals of Mathematics, 2 ser 154:3 (2001) pp.529–587 [1] 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 MR2104929
• 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