Hasse invariant of a quadratic form


In mathematics, the Hasse invariant of a quadratic form Q over a field K takes values in the Brauer group Br. The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.
The quadratic form Q may be taken as a diagonal form
Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras
This is independent of the diagonal form chosen to compute it.
It may also be viewed as the second Stiefel–Whitney class of Q.

Symbols

The invariant may be computed for a specific symbol φ taking values ±1 in the group C2.
In the context of quadratic forms over a local field, the Hasse invariant may be defined using the Hilbert symbol, the unique symbol taking values in C2. The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.
For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.