U-invariant
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the universal invariant or ''u''-invariant of a field 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 to a ...
s 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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, or that every form of dimension at least ''u'' is universal.


Examples

* For the complex numbers, ''u''(C) = 1. * If ''F'' is quadratically closed then ''u''(''F'') = 1. * The function field of an algebraic curve 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 . Examples As an example, the field of real numbers is not algebraically closed, because ...
has ''u'' ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed.Lam (2005) p.376 * If ''F'' is a non-real global or local field, or more generally a linked field, 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 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 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 mathemati ...
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' 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 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.


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