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

Abstract

Let k be a field, G a totally ordered abelian group. The maximal field of generalised power series K=k((G)), endowed with the canonical valuation v, plays a fundamental role in the classification of valued fields (Kaplansky, 1942 and 1945). In this paper we study the group v-AutK of valuation preserving automorphisms of any subfield k(G)⊆K⊆K, where k(G) is the fraction field of the group ring k[G]. Under the assumption that K satisfies two lifting properties we prove a structure theorem decomposing v-AutK into a 4-factor semi-direct product of notable subgroups. We identify a large class of fields satisfying the two aforementioned lifting properties. We then 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 Hom(G,k×) of G into k× and Hom(G,1+IK) of G into the group of 1-units of the valuation ring of K. To illustrate the power of our methods, we apply our results to some special cases, such as the field of Laurent series (Schilling, 1944) and that of Puiseux series (Deschamps, 2005).