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{{redirect|Hasse–Witt invariant|the invariant of algebraic curves|Hasse–Witt matrix}}
In [[mathematics]], the '''Hasse invariant''' (or '''Hasse-Witt invariant''') of a [[quadratic form]] ''Q'' over a [[field (mathematics)|field]] ''K'' takes values in the [[Brauer group]] Br(''K''). That is, it is defined by construction [[central simple algebra]]s over ''K'' from ''Q'', and then evaluating an invariant in the Brauer group constructed from those.
In [[mathematics]], the '''Hasse invariant''' (or '''Hasse–Witt invariant''') of a [[quadratic form]] ''Q'' over a [[field (mathematics)|field]] ''K'' takes values in the [[Brauer group]] Br(''K''). The name "Hasse–Witt" comes from [[Helmut Hasse]] and [[Ernst Witt]].


''Q'' for these purposes may be taken as a [[diagonal form]]
The quadratic form ''Q'' may be taken as a [[diagonal form]]


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


In that case the invariant is the product of the classes in the Brauer group of all the [[quaternion algebra]]s
Its invariant is then defined as the product of the classes in the Brauer group of all the [[quaternion algebra]]s


:(''a''<sub>''i''</sub>, ''a''<sub>''j''</sub>) for ''i'' < ''j''.
:(''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>
This invariant is named for [[Helmut Hasse]] and [[Ernst Witt]].

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]], 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 of a quadratic form|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 [[Signature of a quadratic form|signature]]s 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 | authorlink=Tsit Yuen Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
* {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[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=[[Springer-Verlag]] | isbn=3-540-66564-1 | year=1973 | zbl=0259.10018 }}
* {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=A Course in Arithmetic | series=[[Graduate Texts in Mathematics]] | volume=7 | publisher=[[Springer-Verlag]] | year=1973 | isbn=0-387-90040-3 | zbl=0256.12001 | url-access=registration | url=https://archive.org/details/courseinarithmet00serr }}


[[Category:Quadratic forms]]
[[Category:Quadratic forms]]

Latest revision as of 23:51, 29 October 2024

"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 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

[edit]

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

[edit]

References

[edit]
  1. ^ Lam (2005) p.118
  2. ^ Milnor & Husemoller (1973) p.79
  3. ^ Serre (1973) p.36
  4. ^ Serre (1973) p.39
  5. ^ 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.
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