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).[1]:69

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.[2] The quaternion algebras are linked if and only if the Albert form is isotropic.[1]:70

Linked fields

The field F is linked if any two quaternion algebras over F are linked.[1]: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:[1]: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.[1]:406

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References
  1. Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
  2. Knus, Max-Albert (1991). Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften. 294. Berlin etc.: Springer-Verlag. p. 192. ISBN 3-540-52117-8. Zbl 0756.11008.
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