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
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
- ↑ Lam (2005) p.69
- ↑ 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.
- ↑ Lam (2005) p.70
- ↑ Lam (2005) p.370
- ↑ Lam (2005) p.342
- ↑ 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.