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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

q = ⟨ − a 1 , − a 2 , a 1 a 2 , b 1 , b 2 , − b 1 b 2 ⟩   . {\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. 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:

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

References

  1. ^ Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
  2. Knus, Max-Albert (1991). Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften. Vol. 294. Berlin etc.: Springer-Verlag. p. 192. ISBN 3-540-52117-8. Zbl 0756.11008.
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