Generalized Analysis of Division Algebras of Dimension 2

The mathematical properties of division algebras of dimension 2 are investigated on the basis of the analysis of possible values of the parameters introduced into the laws of composition of basic elements. Generalized expressions for calculating the inverse and neutral elements of the indicated algebras are given. The relations of the parameters defining the normalized division algebras are determined. Possibilities of application of linear orthogonal transformations for the analysis of isomorphism of such algebras are considered. The concept of an exponential function is introduced to represent the elements of the considered non-commutative division algebra in exponential form.

The Norm of a Skew Polynomial

Let D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

Duplication methods for embeddings of real division algebras

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every [Formula: see text]-dimensional (4D) real unital algebra [Formula: see text] into an 8D real unital algebra denoted by [Formula: see text] We also find the conditions on [Formula: see text] under which [Formula: see text] is a division algebra. This covers the most classes of known [Formula: see text]D real division algebras. The second process allows us to embed particular classes of [Formula: see text]D RDAs into [Formula: see text]D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic [Formula: see text]D real division algebras whose derivation algebras contain [Formula: see text]

On Crossed Product Algebras over Henselian Valued Fields

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that Mn([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if Mn(D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with ΓK ⊆ ΓD, where ΓK and ΓD are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.

Factoring octonion polynomials

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree [Formula: see text] with [Formula: see text] conjugacy classes of roots, counting multiplicities.