Hasse Invariant of A Quadratic Form

Hasse Invariant Of A Quadratic Form

In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.

The quadratic form Q may be taken as a diagonal form

Σ a_ix_i2.

Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras

(a_i, a_j) for i < j.

This is independent of the diagonal form chosen to compute it.

It may also be viewed as the second Stiefel–Whitney class of Q.