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) 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.
Properties.
In the case of quadratic extensions, the "u"-invariant is bounded by
and all values in this range are achieved.
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 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 ∞.