In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms
⟨ ⟨ a 1 , a 2 , … , a n ⟩ ⟩ ≅ ⟨ 1 , − a 1 ⟩ ⊗ ⟨ 1 , − a 2 ⟩ ⊗ ⋯ ⊗ ⟨ 1 , − a n ⟩ , {\displaystyle \langle \!\langle a_{1},a_{2},\ldots ,a_{n}\rangle \!\rangle \cong \langle 1,-a_{1}\rangle \otimes \langle 1,-a_{2}\rangle \otimes \cdots \otimes \langle 1,-a_{n}\rangle ,}
for some nonzero elements a1, ..., an of F. (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.) An n-fold Pfister form can also be constructed inductively from an (n−1)-fold Pfister form q and a nonzero element a of F, as q ⊕ ( − a ) q {\displaystyle q\oplus (-a)q} . So the 1-fold and 2-fold Pfister forms look like:
⟨ ⟨ a ⟩ ⟩ ≅ ⟨ 1 , − a ⟩ = x 2 − a y 2 {\displaystyle \langle \!\langle a\rangle \!\rangle \cong \langle 1,-a\rangle =x^{2}-ay^{2}} .
⟨ ⟨ a , b ⟩ ⟩ ≅ ⟨ 1 , − a , − b , a b ⟩ = x 2 − a y 2 − b z 2 + a b w 2 . {\displaystyle \langle \!\langle a,b\rangle \!\rangle \cong \langle 1,-a,-b,ab\rangle =x^{2}-ay^{2}-bz^{2}+abw^{2}.}
For n ≤ 3, the n-fold Pfister forms are norm forms of composition algebras. In that case, two n-fold Pfister forms are isomorphic if and only if the corresponding composition algebras are isomorphic. In particular, this gives the classification of octonion algebras. The n-fold Pfister forms additively generate the n-th power I n of the fundamental ideal of the Witt ring of F.
Characterizations A quadratic form q over a field F is multiplicative if, for vectors of indeterminates x and y, we can write q(x).q(y) = q(z) for some vector z 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. For n-fold Pfister forms with n ≤ 3, this had been known since the 19th century; in that case z can be taken to be bilinear in x and y, by the properties of composition algebras. It was a remarkable discovery by Pfister that n-fold Pfister forms for all n are multiplicative in the more general sense here, involving rational functions. For example, he deduced that for any field F and any natural number n, the set of sums of 2n squares in F is closed under multiplication, using that the quadratic form
x 1 2 + ⋯ + x 2 n 2 {\displaystyle x_{1}^{2}+\cdots +x_{2^{n}}^{2}}
is an n-fold Pfister form (namely, ⟨ ⟨ − 1 , … , − 1 ⟩ ⟩ {\displaystyle \langle \!\langle -1,\ldots ,-1\rangle \!\rangle } ). Another striking feature of Pfister forms is that every isotropic Pfister form is in fact hyperbolic, that is, isomorphic to a direct sum of copies of the hyperbolic plane ⟨ 1 , − 1 ⟩ {\displaystyle \langle 1,-1\rangle } . This property also characterizes Pfister forms, as follows: If q is an anisotropic quadratic form over a field F, and if q becomes hyperbolic over every extension field E such that q becomes isotropic over E, then q is isomorphic to aφ for some nonzero a in F and some Pfister form φ over F.
Connection with K-theory Let kn(F) be the n-th Milnor K-group modulo 2. There is a homomorphism from kn(F) to the quotient In/In+1 in the Witt ring of F, given by
{ a 1 , … , a n } ↦ ⟨ ⟨ a 1 , a 2 , … , a n ⟩ ⟩ , {\displaystyle \{a_{1},\ldots ,a_{n}\}\mapsto \langle \!\langle a_{1},a_{2},\ldots ,a_{n}\rangle \!\rangle ,}
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