In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form ''Q'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' takes values in the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

Br(''K''). The name "Hasse–Witt" comes from Helmut Hasse and

Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...

. The quadratic form ''Q'' may be taken as a

diagonal form In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is :\sum_^n a_i ^m\ for some given degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = ...

:Σ ''a''_''i''''x''_''i''². Its invariant is then defined as the product of the classes in the Brauer group of all the

quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...

s :(''a''_''i'', ''a''_''j'') for ''i'' < ''j''. This is independent of the diagonal form chosen to compute it.Lam (2005) p.118 It may also be viewed as the second

Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...

of ''Q''.

Symbols

The invariant may be computed for a specific symbol φ taking values in the group C₂ = .Milnor & Husemoller (1973) p.79 In the context of quadratic forms over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

, the Hasse invariant may be defined using the

Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity ...

, the unique symbol taking values in C₂.Serre (1973) p.36 The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.Serre (1973) p.39 For quadratic forms over a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

, there is a Hasse invariant ±1 for every

finite place Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

s coming from real embeddings.Conner & Perlis (1984) p.16

References

* * * * * {{cite book , first=Jean-Pierre , last=Serre , authorlink=Jean-Pierre Serre , title=A Course in Arithmetic , series=

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...

, volume=7 , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

, year=1973 , isbn=0-387-90040-3 , zbl=0256.12001 , url-access=registration , url=https://archive.org/details/courseinarithmet00serr Quadratic forms

Symbols

See also

References