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An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup. [ABSTRACT FROM AUTHOR]

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms. [ABSTRACT FROM AUTHOR]

The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 10 to 9. [ABSTRACT FROM AUTHOR]

We extend, to the case of local fields of even characteristic, our previous computations for the set of maximal orders containing given quaternions as concrete sub-graphs of the Bruhat–Tits tree. Computing the relative position of such sub-graphs, the branches, is useful, for instance, to explicitly compute relative spinor images, thus solving the selectivity problem. They also play a role in the description of quotient graphs, which are useful to study matrix groups of arithmetical importance. As in earlier work, we concentrate on orders generated by two quaternions, but extending our method to larger generating sets is quite straightforward. In our previous work, the results were given mainly in terms of the quadratic defect. In the present context, we introduce and characterize an analogous concept for Artin–Scheier extensions, to take care of quaternions generating Galois field extensions. The latter extensions have no non-trivial elements of null trace. For this reason, we state our results in terms of a wider family of algebra presentations, in contrast to the aforementioned work, where we chose a pair of pure quaternions as generators of the order. [ABSTRACT FROM AUTHOR]