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 (forms)|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 [quadratic form|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:F is linked.
- Any two quaternion algebras over F are linked.
- Every Albert form 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.