In mathematics, the universal invariant or ''u''-invariant of a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

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 In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...

over ''F'', or ∞ if this does not exist. Since

formally real field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above i ...

s 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 Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...

Examples

* For the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s, ''u''(C) = 1. * If ''F'' is quadratically closed then ''u''(''F'') = 1. * The function field of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

over an algebraically closed field has ''u'' ≤ 2; this follows from Tsen's theorem that such a field is

quasi-algebraically closed In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebra ...

.Lam (2005) p.376 * If ''F'' is a non-real

global Global means of or referring to a globe and may also refer to: Entertainment * ''Global'' (Paul van Dyk album), 2003 * ''Global'' (Bunji Garlin album), 2007 * ''Global'' (Humanoid album), 1989 * ''Global'' (Todd Rundgren album), 2015 * Bruno ...

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

, or more generally a

linked field In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. Linked quaternion algebras Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B ...

, then ''u''(''F'') = 1, 2, 4 or 8.Lam (2005) p.406

Properties

* If ''F'' is not formally real and the characteristic of ''F'' is not ''2'' then ''u''(''F'') is at most

q(F) = \left, \

, the index of the squares in the multiplicative

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of ''F''.Lam (2005) p. 400 * ''u''(''F'') cannot take the values 3, 5, or 7.Lam (2005) p. 401 Fields exist with ''u'' = 6Lam (2005) p.484 and ''u'' = 9. * Merkurjev has shown that every

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...

integer occurs as the value of ''u''(''F'') for some ''F''.Lam (2005) p. 402Elman, Karpenko, Merkurjev (2008) p. 170 *

Alexander Vishik Alexander is a male given name. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here are Aleksandar, Al ...

proved that there are fields with ''u''-invariant

2^r+1

for all

r > 3

. * The ''u''-invariant is bounded under finite-

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

field extensions. If ''E''/''F'' is a field extension of degree ''n'' then ::

u(E) \le \frac u(F) \ .

In the case of quadratic extensions, the ''u''-invariant is bounded by :

u(F) - 2 \le u(E) \le \frac u(F) \

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 In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...

of the Witt ring of F, or ∞ if this does not exist.Lam (2005) p. 409 For non-formally-real fields, the Witt ring is torsion, so this agrees with the previous definition.Lam (2005) p. 410 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 In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...

References

* * * {{cite book , title=The algebraic and geometric theory of quadratic forms , volume=56 , series=American Mathematical Society Colloquium Publications , first1=Richard , last1=Elman , first2=Nikita , last2=Karpenko , first3=Alexander , last3=Merkurjev , publisher=American Mathematical Society, Providence, RI , year=2008 , isbn=978-0-8218-4329-1 Field (mathematics) Quadratic forms