On the classification of rational four-dimensional unital division algebras
Volume 162 / 2020
Abstract
In a 2019 paper by E. Dieterich, the category $\mathscr {C}(k)$ 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.
