The automorphism group of a valued field of generalised formal power series

TL;DR

This paper investigates the structure of valuation-preserving automorphisms of generalized power series fields, providing a decomposition theorem and explicit descriptions for automorphism groups, extending previous results on Laurent and Puiseux series.

Contribution

It generalizes Hofberger's decomposition of automorphism groups and offers a detailed structure theorem for valuation-preserving automorphisms in generalized power series fields.

Findings

Decomposition of automorphism group into a 4-factor semi-direct product.

Explicit description of strongly additive automorphisms.

Extension of results to fields like Laurent and Puiseux series.

Abstract

Let $ k $ be a field, $ G $ a totally ordered abelian group and $ \mathbb K = k((G)) $ the maximal field of generalised power series, endowed with the canonical valuation $ v $. We study the group $ v \mathrm{-Aut} K $ of valuation preserving automorphisms of a subfield $ k(G)\subseteq K\subseteq \mathbb K $, where $ k(G) $ is the fraction field of the group ring $ k[G] $. Under the assumption that $ K $ satisfies two lifting properties we are able to generalise and refine Hofberger's decomposition of $ v \mathrm{-Aut}\mathbb K $ and prove a structure theorem decomposing $ v\mathrm{-Aut} K $ into a 4-factor semi-direct product of notable subgroups. We then identify a large class of Hahn fields satisfying the two aforementioned lifting properties. Next we focus on the group of strongly additive automorphisms of $ K $. We give an explicit description of the group of strongly additive internal automorphisms in terms of the groups of homomorphisms $ \mathrm{Hom}(G,k^\times) $ of $ G $ into $ k^\times $ and $ \mathrm{Hom}(G,1+I_K) $ of $ G $ into the group of 1-units of the valuation ring of $ K $. Finally, we specialise our results to some relevant special cases. In particular, we extend the work of Schilling on the field of Laurent series and that of Deschamps on the field of Puiseux series.