Algebraic structures on non-Archimedean Urysohn universal metric spaces

Abstract

We investigate valued-field structures on Urysohn universal ultrametric spaces. We introduce $p$-adic Levi--Civita fields as subfields of $p$-adic Hahn fields and treat them together with ordinary Levi--Civita fields. For a subgroup $G$ of $\mathbb{R}$ containing $\mathbb{Z}$ and a countably infinite perfect field $k$, the corresponding Levi--Civita valued field is isometric to the $R$-Urysohn universal ultrametric space, where $R=\{0\}\cup\{η^{-g}\mid g\in G\}$. Thus these spaces admit field structures extending prescribed prime valued fields, including $\mathbb{Q}$ with the trivial valuation and the $p$-adic fields $\mathbb{Q}_{p}$. We also prove that complete valued fields with infinite residue fields are haloed, and hence universal for separable ultrametric spaces with corresponding distance sets. In the separable case, such a valued field is itself isometric to the corresponding Urysohn space. Examples include $\mathbb{C}_{p}$, the completion of the maximal unramified extension of $\mathbb{Q}_{p}$, Laurent series fields, and completions of their algebraic closures. Finally, for a countably infinite perfect residue field, the corresponding full Hahn-type valued field is a Urysohn universal ultrametric space exactly when its value group is order-isomorphic to $\mathbb{Z}$.