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").
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.