On the classification of rational four-dimensional unital division algebras
Volume 162 / 2020
Abstract
In a 2019 paper by E. Dieterich, the category of four-dimensional unital division algebras whose right nucleus is non-trivial and whose automorphism group contains Klein’s four-group V is studied over a general ground field k with \mathrm{char}\, k\neq 2. In particular, the objects in \mathscr {C}(k) are exhaustively constructed from parameters in k^3, and explicit isomorphism conditions for the objects constructed are found in terms of these parameters.
In this paper, we specialize to the case k=\mathbb {Q} and present results towards a classification of \mathscr {C}(\mathbb {Q}). In particular, for each field \ell with [\ell :k]=2 we present an explicit two-parameter family of pairwise non-isomorphic non-associative objects in \mathscr {C}(\mathbb {Q}) that admit \ell as a subfield and we provide a method for classifying the full subcategory formed by all central skew fields over \mathbb {Q} that admit \ell as a subfield and kV-submodule. We also classify the full subcategory of \mathscr {C}(\mathbb {Q}) formed by all Galois extensions of \mathbb {Q} with Galois group V that admit \ell as a subfield.