mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

s over the field. The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an

anisotropic quadratic space In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms and ...

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 ...

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 In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...

, 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 that is a subsidiary of Comcast ** Universal Animation Studios, an American Animation studio, and a subsidiary of N ...

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 for ...

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 cu ...

over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

has ''u'' ≤ 2; this follows from

Tsen's theorem In mathematics, Tsen's theorem states that a function field ''K'' of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally t ...

that such a field is

quasi-algebraically closed In mathematics, a field ''F'' is called quasi-algebraically closed (or ''C''1) 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-alg ...

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

global Global may refer to: General *Globe, a spherical model of celestial bodies *Earth, the third planet from the Sun Entertainment * ''Global'' (Paul van Dyk album), 2003 * ''Global'' (Bunji Garlin album), 2007 * ''Global'' (Humanoid album), 198 ...

local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

, 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 iden ...

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), a Breton surname * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a ...

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 name of Greek origin. 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 A ...

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

2^r+1

for all

r > 3

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

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

s. 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 ...

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 a Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \la ...

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