In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.

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