In mathematics, a linked field is a
field for which the
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 ...
s 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'' = (''b''
1,''b''
2) be quaternion algebras over ''F''. The algebras ''A'' and ''B'' are linked quaternion algebras over ''F'' if there is ''x'' in ''F'' such that ''A'' is equivalent to (''x'',''y'') and ''B'' is equivalent to (''x'',''z'').
The Albert form for ''A'', ''B'' is
:
It can be regarded as the difference in the
Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''.
The quaternion algebras are linked if and only if the Albert form is
isotropic.
[
]
Linked fields
The field ''F'' is ''linked'' if any two quaternion algebras over ''F'' are linked.[ Every ]global
Global means of or referring to a globe and may also refer to:
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and 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 compa ...
is linked since all quadratic forms of degree 6 over such fields are isotropic.
The following properties of ''F'' are equivalent:[
* ''F'' is linked.
* Any two quaternion algebras over ''F'' are linked.
* Every ''Albert form'' (dimension six form of discriminant −1) is isotropic.
* The quaternion algebras form a subgroup of the ]Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:-
* Alfred Brauer (1894–1985), German-American mathematician, brother of Richard
* Andreas Brauer (born 1973), German film producer
* Arik ...
of ''F''.
* Every dimension five form over ''F'' is a Pfister neighbour In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold Pf ...
.
* No biquaternion algebra In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field.
The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense.
...
over ''F'' is a division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a f ...
.
A nonreal linked field has 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''(''F'') of a field ''F'' is the largest dimension of an anisotropic quadratic space ove ...
equal to 1,2,4 or 8.[
]
References
* {{cite journal , last=Gentile , first=Enzo R. , title=On linked fields , journal=Revista de la Unión Matemática Argentina , volume=35 , pages=67–81 , year=1989 , url=http://inmabb.criba.edu.ar/revuma/pdf/v35/p067-081.pdf , issn=0041-6932 , zbl=0823.11010
Field (mathematics)
Quadratic forms
Quaternions