Small extensions of analytic fields
Michael Temkin
TL;DR
The paper proves that the topological generating degree ${\rm Tr.c}$ is monotone under passage to subextensions for small analytic field extensions, and it characterizes the intermediate algebraically closed subfields in key one-generator cases via a distance parameter. It establishes a natural piecewise-linear (PL) structure on the interval of intermediate fields $I_{K/k}$ when $K=\widehat{k(t)^a}$, with transition functions that are $(p^{\mathbb Z},|K^\times|)$-PL and a bijection $\rho_t:I_{K/k}\to[0,r_k(t)]$ encoding the radii. The work also develops a comprehensive geometric interpretation of fields of definition, radii, and metrics on Berkovich-type spaces, and extends the analysis to large extensions with finite topological transcendence degree, highlighting cofinality phenomena and posing further questions. Together these results provide a detailed, quantitative picture of how intermediate fields sit inside small and large extensions, and how radii and PL transitions control their structure and automorphisms. The findings have implications for Berkovich and diamonds frameworks and deepen understanding of independence, generation, and ramification in non-archimedean analytic geometry.
Abstract
An extension $K/k$ of analytic (i.e. real valued complete) fields is called small if it is topologically-algebraically generated by finitely many elements. We prove that this property is inherited by subextensions and hence topological generating degree of such extensions is monotonic. Much more detailed results are obtained in the case of degree one. Let $k$ be an analytic algebraically closed field of positive residual characteristic $p$ and $K=\widehat{k(t)^a}$ with a non-trivial valuation. In a previous work it was shown that the set $I_{K/k}$ of intermediate complete algebraically closed subextensions $k\subseteq F\subseteq K$ is totally ordered by inclusion. In this paper we show that $I_{K/k}$ is an interval parameterized by the distance between $t$ and $F$. Moreover, logarithmic parameterizations induced by other generators differ by PL functions with slopes in $p^{\mathbb Z}$ and corners in $|K^\times|$, so $I_{K/k}$ acquires a natural PL structure.