Discretely valued ternary fields and Hahn ternary fields over \(\mathbb{Z}\) (Q5954175)

Let \(N=(N,v,{\mathbb Z}_{-\infty})\) be a discretely valued ternary field as introduced by \textit{F. Kalhoff} [Geom. Dedicata 28, 337-348 (1988; Zbl 0665.51001)]. \textit{S. Prieß-Crampe} and \textit{P. Ribenboim} [J. Algebra 186, No. 2, 401-435 (1996; Zbl 0866.54027)] compared such a ternary field \(N\) with a Hahn ternary field \(H=H({\mathbb Z}, N_v, {\mathcal C})\), where \(N_v = N / \ker v\) is the residue ternary field and \({\mathcal C}\) is some suitable factor system, and searched for criteria ensuring the existence of an isometric embedding \(\phi : (N,v,{\mathbb Z}_{-\infty}) \to (H,v_H,{\mathbb Z}_{-\infty})\). If the image \(\phi(N)\) contains both sets \(N_vt^0\) and \([1]t^{\mathbb Z}\), then the embedding \(\phi\) is called regular. Based on some of his earlier work [Geom. Dedicata 80, No. 1-3, 157-171 (2000; Zbl 0959.51003) and Geom. Dedicata 80, No. 1-3, 211-230 (2000; Zbl 0958.51003)] the author derives a necessary and sufficient condition for the existence of a regular embedding as explained before. This result is then applied to richer algebraic structures such as Cartesian groups and division algebras.