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).
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: 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.
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