U-invariant

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) or 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

• For the complex numbers, u(C) = 1.
• If F is 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.

Properties

• If F is not formally real then u(F) is at most $q(F)=\left|{F^{\star }/F^{\star 2}}\right|$, the index of the 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 ∞ if this does 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 ∞.

Properties

• u(F) ≤ 1 if and only if F is a Pythagorean field.

References

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.