Hasse invariant of a quadratic form

"Hasse–Witt invariant" redirects here. For the invariant of algebraic curves, see Hasse–Witt matrix.

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.^[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 ±1 in the group C₂.^[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]

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. 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. 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. 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. 117. Springer-Verlag. ISBN 3-540-66564-1. Zbl 0259.10018.
Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.

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