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Linked field

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In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.

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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).

The Albert form for A, B is

q = ⟨ − a 1 , − a 2 , a 1 a 2 , b 1 , b 2 , − b 1 b 2 ⟩   .

It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if the Albert form is isotropic.

Linked fields

The field F is linked if any two quaternion algebras over F are linked. 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:

  • 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.

    References

    Linked field Wikipedia
    (Text) CC BY-SA