# Linked field

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In mathematics, a '''linked field''' is a [field](/source/Field_(algebra)) for which the [quadratic form](/source/quadratic_form)s attached to [quaternion algebra](/source/quaternion_algebra)s have a common property.

==Linked quaternion algebras==
Let ''F'' be a field of [characteristic](/source/Characteristic_(algebra)) not equal to 2.  Let ''A'' = (''a''<sub>1</sub>,''a''<sub>2</sub>) and ''B'' = (''b''<sub>1</sub>,''b''<sub>2</sub>) 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'').<ref name=Lam>{{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[Graduate Studies in Mathematics](/source/Graduate_Studies_in_Mathematics) | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=[American Mathematical Society](/source/American_Mathematical_Society) | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}</ref>{{rp|69}}

The '''Albert form''' for ''A'', ''B'' is

:<math>q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . </math>

It can be regarded as the difference in the [Witt ring](/source/Witt_ring_(forms)) of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''.<ref>{{cite book | last=Knus | first=Max-Albert |authorlink=Max-Albert Knus| title=Quadratic and Hermitian forms over Rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[Springer-Verlag](/source/Springer-Verlag) | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=192 }}</ref>  The quaternion algebras are linked if and only if the Albert form is [isotropic](/source/isotropic_quadratic_form).<ref name=Lam/>{{rp|70}}

==Linked fields==
The field ''F'' is ''linked'' if any two quaternion algebras over ''F'' are linked.<ref name=Lam/>{{rp|370}}  Every [global](/source/global_field) and [local field](/source/local_field) is linked since all quadratic forms of degree 6 over such fields are isotropic.

The following properties of ''F'' are equivalent:<ref name=Lam/>{{rp|342}}
* ''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](/source/Brauer_group) of ''F''.
* Every dimension five form over ''F'' is a [Pfister neighbour](/source/Pfister_neighbour).
* No [biquaternion algebra](/source/biquaternion_algebra) over ''F'' is a [division algebra](/source/division_algebra).

A nonreal linked field has [u-invariant](/source/u-invariant) equal to 1,2,4 or 8.<ref name=Lam/>{{rp|406}}

==References==
{{reflist}}

* {{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 }}

Category:Field theory
Category:Quadratic forms
Category:Quaternions

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