Linked field
# 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 $(x,z).^{:69}$

The Albert form for A, B is

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

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if the Albert form is $isotropic.^{:70}$

## Linked fields

The field F is linked if any two quaternion algebras over F are $linked.^{:370}$ 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 $equivalent:^{:342}$

* 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 $8.^{:406}$

