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

${\displaystyle q=\left\langle {-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle \ .}$

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