View source for Hasse invariant of a quadratic form

In [[Mathematics|mathematics]], the '''Hasse invariant''' (or '''Hasse–Witt invariant''') of a [[Quadratic form|quadratic form]] ''Q'' over a [[Field (mathematics)|field]] ''K'' takes values in the [[Brauer group]] Br(''K'').  The name "Hasse–Witt" comes from [[Biography:Helmut Hasse|Helmut Hasse]] and [[Biography:Ernst Witt|Ernst Witt]].

The quadratic form ''Q''  may be taken as a [[Diagonal form|diagonal form]]

:Σ ''a''<sub>''i''</sub>''x''<sub>''i''</sub><sup>2</sup>.

Its invariant is then defined as the product of the classes in the Brauer group of all the [[Quaternion algebra|quaternion algebra]]s

:(''a''<sub>''i''</sub>, ''a''<sub>''j''</sub>) for ''i'' < ''j''.

This is independent of the diagonal form chosen to compute it.<ref name=Lam118>Lam (2005) p.118</ref>

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

==Symbols==
The invariant may be computed for a specific [[Steinberg symbol|symbol]] φ  taking values in the group C<sub>2</sub> = {±1}.<ref name=MH79>Milnor & Husemoller (1973) p.79</ref>

In the context of quadratic forms over a [[Local field|local field]], the Hasse invariant may be defined using the [[Hilbert symbol]], the unique symbol taking values in C<sub>2</sub>.<ref name=S36>Serre (1973) p.36</ref>  The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.<ref name=S39>Serre (1973) p.39</ref>

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.<ref name=CP16>Conner & Perlis (1984) p.16</ref>

== See also ==

* Hasse–Minkowski theorem

==References==
{{reflist}}
* {{cite book | first1=P.E. | last1=Conner | first2=R. | last2=Perlis | title=A Survey of Trace Forms of Algebraic Number Fields | series=Series in Pure Mathematics | volume=2 | publisher=World Scientific | year=1984 | isbn=9971-966-05-0 | zbl=0551.10017 }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[[Graduate Studies in Mathematics]] | first=Tsit-Yuen | last=Lam |  publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
* {{cite book | first1=J. | last1=Milnor |  first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=Ergebnisse der Mathematik und ihrer Grenzgebiete | volume=73 | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 }}
* {{cite book | last=O'Meara | first=O.T. | title=Introduction to quadratic forms | series=Die Grundlehren der mathematischen Wissenschaften | volume=117 | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | isbn=3-540-66564-1 | year=1973 | zbl=0259.10018 }}
* {{cite book | first=Jean-Pierre | last=Serre |  title=A Course in Arithmetic | series=[[Graduate Texts in Mathematics]] | volume=7 | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | year=1973 | isbn=0-387-90040-3 | zbl=0256.12001 | url-access=registration | url=https://archive.org/details/courseinarithmet00serr }}

[[Category:Quadratic forms]]

{{Sourceattribution|Hasse invariant of a quadratic form}}