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 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 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.

Examples

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 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