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