|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''Pfister form''' is a particular kind of [[quadratic form]] over a [[field (mathematics)|field]] ''F'' (whose [[characteristic (algebra)|characteristic]] is usually assumed to be not 2), introduced by [[Albrecht Pfister (mathematician)|Albrecht Pfister]] in 1965. A Pfister form is in 2<sup>''n''</sup> variables, for some natural number ''n'' (also called an '''n-Pfister form'''), and may be written as a [[tensor product of quadratic forms]] as:
| | Greetings! I am Myrtle Shroyer. Bookkeeping is what I do. To gather badges is what her family members and her appreciate. Her husband and her reside in Puerto Rico but she will have to move one working day or another.<br><br>My web blog; [http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw] |
| | |
| :<math>\langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle \cong \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle,</math>
| |
| | |
| for ''a<sub>i</sub>'' elements of the field ''F''. An ''n''-Pfister form may also be constructed inductively from an (''n''-1)-Pfister form ''q'' and an element ''a'' of ''F'', as <math>q \oplus (a)q</math>.
| |
| | |
| So all 1-Pfister forms and 2-Pfister forms look like:
| |
| | |
| :<math>\langle\!\langle a\rangle\!\rangle\cong \langle 1, a \rangle \cong x^2 + ay^2</math>.
| |
| :<math>\langle\!\langle a,b\rangle\!\rangle\cong \langle 1, a, b, ab \rangle \cong x^2 + ay^2 +bz^2 +abw^2.</math>
| |
| | |
| For n ≤ 3 the ''n''-Pfister forms are [[norm form]]s of [[composition algebra]]s.<ref name=Lam316>Lam (2005) p.316</ref> In fact, in this case, two ''n''-Pfister forms are [[Isometry|isometric]] if and only if the corresponding composition algebras are [[isomorphic]].
| |
| | |
| The Pfister forms are generators for the torsion in the [[Witt group]].<ref name=Lam395>Lam (2005) p.395</ref> The ''n''-fold forms additively generate the ''n''-th power ''I''<sup>''n''</sup> of the fundamental ideal of the Witt ring.<ref name=Lam316>Lam (2005) p.316</ref>
| |
| | |
| ==Characterisation==
| |
| We define a quadratic form ''q'' over a field ''F'' to be '''multiplicative''' if when '''x''' and '''y''' are vectors of indeterminates, then ''q''('''x''').''q''('''y''') = ''q''('''z''') where '''z''' is a vector of [[rational function]]s in the '''x''' and '''y''' over ''F''. [[Isotropic quadratic form]]s are multiplicative.<ref name=Lam324>Lam (2005) p.324</ref> For [[anisotropic quadratic form]]s, Pfister forms are multiplicative and conversely.<ref name=Lam325>Lam (2005) p.325</ref><ref name=Raj164>Rajwade (1993) p.164</ref>
| |
| | |
| ==Connection with K-theory==
| |
| Let ''k''<sub>''n''</sub>(''F'') be the ''n''-th group in [[Milnor K-theory]] modulo 2. There are homomorphisms from ''k''<sub>''n''</sub>(''F'') to the Witt ring by taking the symbol
| |
| | |
| :<math> \{a_1,\ldots,a_n\} \mapsto \langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle ,</math> | |
| | |
| where the image is an ''n''-fold Pfister form</sup>.<ref name=Lam366>Lam (2005) p.366</ref> The image can be taken as ''I''<sup>''n''</sup>/''I''<sup>''n''+1</sup> and the map is surjective since the Pfister forms additively generate ''I''<sup>''n''</sup>. The [[Milnor conjecture]] can be interpreted as stating that these maps are isomorphisms.<ref name=Lam366/>
| |
| | |
| ==Pfister neighbours==
| |
| A '''Pfister neighbour''' is a form (''W'',σ) such that (''W'',σ) is similar to a subspace of a space with Pfister form (''V'',φ) where dim.''V'' < 2 dim.''W''.<ref name=Lam339>Lam (2005) p.339</ref> The associated Pfister form φ is uniquely determined by σ. Any ternary form is a Pfister neighbour; a quaternary form is a Pfister neighbour if and only if its discriminant is a square.<ref name=Lam341>Lam (2005) p.341</ref> A degree five form is a Pfister neighbour if and only if the underlying field is a [[linked field]].<ref name=Lam342>Lam (2005) p.342</ref>
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{Citation | 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 }}, Ch. 10
| |
| * {{citation | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
| |
| | |
| ==Further reading==
| |
| * {{citation | last1=Knebusch | first1=Manfred | last2=Scharlau | first2=Winfried | title=Algebraic theory of quadratic forms. Generic methods and Pfister forms | others=Notes taken by Heisook Lee | series=DMV Seminar | volume=1 | location=Boston - Basel - Stuttgart | publisher=Birkhäuser Verlag | year=1980 | isbn=3-7643-1206-8 | zbl=0439.10011 }}
| |
| | |
| [[Category:Quadratic forms]]
| |
Greetings! I am Myrtle Shroyer. Bookkeeping is what I do. To gather badges is what her family members and her appreciate. Her husband and her reside in Puerto Rico but she will have to move one working day or another.
My web blog; http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw