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
- [math]\displaystyle{ q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . }[/math]
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
- ↑ 1.0 1.1 1.2 1.3 1.4 Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2.
- ↑ 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.
- Gentile, Enzo R. (1989). "On linked fields". Revista de la Unión Matemática Argentina 35: 67–81. ISSN 0041-6932. http://inmabb.criba.edu.ar/revuma/pdf/v35/p067-081.pdf.
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