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]

The Albert form for A, B is

$q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ .$

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]

Linked fields

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]

• 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.^[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

Bibliography

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

Further reading

• Gentile, Enzo R. (1989). "On linked fields". Rev. Unión Mat. Argent. 35: 67–81. ISSN 0041-6932. Zbl 0823.11010.