In
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
, 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'' = 6
[Lam (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. 402][Elman, 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
for all
.
* 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
::
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.
[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