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'' = (''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'').<ref name=Lam>{{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=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 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 | 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.<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 and 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 of ''F''. * Every dimension five form over ''F'' is a Pfister neighbour. * No biquaternion algebra over ''F'' is a division algebra.

A nonreal linked field has 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