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 = and B = 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 and B is equivalent to.
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 and local field is linked since all quadratic forms of degree 6 over such fields are isotropic.
The following properties of F are equivalent:
A nonreal linked field has u-invariant equal to 1,2,4 or 8.