Adam Chapman

Chain equivalences for symplectic bases, quadratic forms and tensor products of quaternion algebras

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.

Linkage of quadratic Pfister forms

ABSTRACT We study the necessary conditions for sets of quadratic n-fold Pfister forms to have a common (n−1)-fold Pfister factor. For any set S of n-fold Pfister forms generating a subgroup of of order 2s in which every element has an n-fold Pfister representative, we associate an invariant in which lives inside when the forms in S have a common (n−1)-fold Pfister factor. We study the properties of this invariant and compute it explicitly in a few interesting cases.

Symbol length of p-algebras of prime exponent

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree

Standard Polynomial Equations Over Division Algebras

p

Tensor products of cyclic algebras of degree 4 and their Kummer subspaces

over a field

Quaternion quadratic equations in characteristic 2

F

General polynomials over division algebras and left eigenvalues

with

Total linkage of quaternion algebras and Pfister forms in characteristic two

\operatorname{char}(F)=p

On the generalized Clifford algebra of a monic polynomial

is a finite integer

Kummer subspaces of tensor products of cyclic algebras

d