Linked field explained

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.^[1]

Linked fields

The field F is linked if any two quaternion algebras over F are linked.^[1] 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:^[1]

• 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.^[1]

References

• Gentile . Enzo R. . On linked fields . Revista de la Unión Matemática Argentina . 35 . 67–81 . 1989 . 0041-6932 . 0823.11010 .

Notes and References

1. Book: Lam, Tsit-Yuen . Introduction to Quadratic Forms over Fields . 67 . . T. Y. Lam . . 2005 . 0-8218-1095-2 . 1068.11023 . 2104929 .
2. Book: Knus, Max-Albert . Quadratic and Hermitian forms over rings . Grundlehren der Mathematischen Wissenschaften . 294 . Berlin etc. . . 1991 . 3-540-52117-8 . 0756.11008 . 192 .