A proof of the Pfister Factor Conjecture

Abstract

It is shown that any split product of quatemion algebras with orthogonal involution is adjoint to a Pfister form. This settles the Pfister Factor Conjecture formulated by D.B. Shapiro. A more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quatemion algebra.Let K be a field of characteristic different from 2. By a K -algebra with involution we mean a pair (A, a) of a central simple K -algebra A and a K -linear involution a : A ---* A. Involutions of this kind are either orthogonal or symplectic.A (regular) quadratic form <p over K can be understood as a pair (V, (', '}",) of a finite-dimensional K-vector space V and a nonsingular symmetricbilinearform(·, ')",: VxV ---* K.LetAd(<p) denote the adjointAssigning to a quadratic form its adjoint algebra with involution yields a one-to-one correspondence between the similarity classes of quadratic forms over K and the isomorphism classes of split K -algebras with orthogonal involution (see [7, p. 1]). This correspondence commutes with taking tensor products and with extension of the field of scalars. The notions of isotropy and hyperbolicity are defined for algebras with involution in such a way that, in particular, a quadratic form is anisotropic, isotropic, or hyperbolic if and only if its adjoint algebra with involution has the same property (see [2]).Since Pfister forms playa key role in the theory of quadratic forms, it seems natural to look for a corresponding concept in the theory of algebras with involution. An n-fold Pfister form over K is a quadratic form of the shape (1, a] }0· .. 0 (1, all) over K, with a], ... ,all E KX. The algebra with involution adjoint to this form is isomorphic to 0;1=] Ad( (l, ai)), which is

Totally decomposable symplectic and unitary involutions

“…We show that for every field extension, totally decomposable symplectic involutions on index two algebras and unitary involutions on split algebras are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of two-power degree. These results were first proven in [3] and [6, (3.1)] respectively under the assumption that the base field was of characteristic different from two. Here we make no assumption on the characteristic of the base field.…”

“…As this holds for any field extension L/F , ρ is similar to a Pfister form by Proposition 3. 3. Similarly, ρ F (π) is isotropic, and hence hyperbolic.…”

“…That totally decomposable orthogonal involutions over a field of characteristic different from two are always either anisotropic or hyperbolic can be shown (see [6, (3.2)]) using the nonhyperbolic splitting results of Karpenko, [10], and that any totally decomposable orthogonal involution on a split algebra is adjoint to a Pfister form. The later result is known as the Pfister Factor Conjecture, and was shown in [3]. The converse, that any orthogonal involution on an algebra of two-power degree that is anisotropic or hyperbolic after extending scalars to any field extension is necessarily totally decomposable, is clear for split algebras, but otherwise remains largely open.…”

“…Our approach follows that used in [3]. The main tool is an index reduction argument using the the so-called trace forms of Jacobson, first introduced for characteristic different from two in [13].…”

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

“…For the case of characteristic = 2, these involutions were studied in [8] in connection with the periodicity of real Clifford algebras with involution. Also they were considered in [15] in connection with the Pfister Factor Conjecture, which was finally settled in [1]. In characteristic = 2, some properties of these involutions were also investigated in [9].…”