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 aixi.
Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras
- (ai, aj) 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.
Symbols
The invariant may be computed for a specific symbol φ taking values in the group C2 = {±1}.
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.
See also
References
- Lam (2005) p.118
- Milnor & Husemoller (1973) p.79
- Serre (1973) p.36
- Serre (1973) p.39
- Conner & Perlis (1984) p.16
- Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
- Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
- Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
- O'Meara, O.T. (1973). Introduction to quadratic forms. Die Grundlehren der mathematischen Wissenschaften. Vol. 117. Springer-Verlag. ISBN 3-540-66564-1. Zbl 0259.10018.
- Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. Vol. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.