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 = (a₁,a₂) and B = (b₁,b₂) 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).{{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 }}{{rp|69}}

The Albert form for A, B is

: $q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ .$

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B.{{cite book | last=Knus | first=Max-Albert | 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 }} The quaternion algebras are linked if and only if the Albert form is isotropic.{{rp|70}}

Linked fields

The field F is linked if any two quaternion algebras over F are linked.{{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:{{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.{{rp|406}}

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 }}

Category:Field (mathematics)

Category:Quadratic forms

Category:Quaternions