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]^:69

The Albert form for A, B is

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

Linked fields

The field F is linked if any two quaternion algebras over F are linked.^[1]^:370 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]^:342

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]^:406

References

1 2 3 4 5 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.
↑ 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.

Gentile, Enzo R. (1989). "On linked fields" (PDF). Revista de la Unión Matemática Argentina. 35: 67–81. ISSN 0041-6932. Zbl 0823.11010.

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[Lam-1] 1 2 3 4 5 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.

[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.