In mathematics, a Pfister form is a particular kind of quadratic form over a field F (whose characteristic is usually assumed to be not 2), introduced by Albrecht Pfister in 1965. A Pfister form is in 2n variables, for some natural number n (also called an n-Pfister form), and may be written as a tensor product of quadratic forms as:
for ai 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 .
So all 1-Pfister forms and 2-Pfister forms look like:
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 functions in the x and y over F. Isotropic quadratic forms are multiplicative. For anisotropic quadratic forms, Pfister forms are multiplicative and conversely.
Connection with K-theory
Let kn(F) be the n-th group in Milnor K-theory modulo 2. There are homomorphisms from kn(F) to the Witt ring by taking the symbol
where the image is an n-fold Pfister form. The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In. The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.
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. 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. A degree five form is a Pfister neighbour if and only if the underlying field is a linked field.
- Lam (2005) p.316
- Lam (2005) p.395
- Lam (2005) p.324
- Lam (2005) p.325
- Rajwade (1993) p.164
- Lam (2005) p.366
- Lam (2005) p.339
- Lam (2005) p.341
- Lam (2005) p.342
- 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, Ch. 10
- Rajwade, A. R. (1993), Squares, London Mathematical Society Lecture Note Series 171, Cambridge University Press, ISBN 0-521-42668-5, Zbl 0785.11022