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

For the complex numbers, u = 1.
If F is quadratically closed then u = 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.
If F is a non-real global or local field, or more generally a linked field, then u = 1, 2, 4 or 8.

Properties

If F is not formally real and the characteristic of F is not 2 then u is at most, the index of the squares in the multiplicative group of F.u cannot take the values 3, 5, or 7. Fields exist with u = 6 and u = 9.
Merkurjev has shown that every even integer occurs as the value of u for some F.
Alexander Vishik proved that there are fields with u-invariant for all.
The u-invariant is bounded under finite-degree field extensions. If E/''F is a field extension of degree n'' then

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 ∞.

Properties

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