On the classification of rational four-dimensional unital division algebras

In a paper by E. Dieterich 2017, 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 constructed objects 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 explicity a 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 of central skew fields admitting $\ell$ as a subfield and $kV$-submodule. We also classify the subcategory of $\mathscr{C}(\mathbb{Q})$ of all four-dimensional Galois extensions of $\mathbb{Q}$ with Galois group $V$ that admit $\ell$ as a subfield.