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

## Contents

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]

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.[3]

The field F is linked if any two quaternion algebras over F are linked.[4] Every global and local field is linked since all quadratic forms of dgree 6 over such fields are isotropic.

The following properties of F are equivalent:[5]

A nonreal linked field has u-invariant equal to 1,2,4 or 8.[6]

## References

1. ^ Lam (2005) p.69
2. ^ Knus, Max-Albert (1991). Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften 294. Berlin etc.: Springer-Verlag. p. 192. ISBN 3-540-52117-8. Zbl 0756.11008.
3. ^ Lam (2005) p.70
4. ^ Lam (2005) p.370
5. ^ Lam (2005) p.342
6. ^ Lam (2005) p.406