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 aixi2.
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.[1]
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}.[2]
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.[3] The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.[4]
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.[5]
See also
- Hasse–Minkowski theorem
References
- Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0.
- Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2.
- Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X.
- O'Meara, O.T. (1973). Introduction to quadratic forms. Die Grundlehren der mathematischen Wissenschaften. 117. Springer-Verlag. ISBN 3-540-66564-1.
- Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Springer-Verlag. ISBN 0-387-90040-3. https://archive.org/details/courseinarithmet00serr.
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