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 [2]. In this talk we study the group 𝓋-Aut 𝐾 of valuation preserving automorphisms of a 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 present a structure theorem decomposing 𝓋-Aut 𝐾 into a 4-factor semi-direct product of notable subgroups. We identify a large class of fields satisfying the two aforementioned lifting properties. 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 Hom(G, 𝑘×) of G into 𝑘× and Hom(G, 1 + Iᴋ ) 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 [4] and that of Puiseux series [1].

References:

[1] B. Deschamps, Des automorphismes continus d’un corps de s´eries de Puiseux, Acta Arith. 118 (3)

(2005) 205–229. DOI:10.4064/aa118-3-1.

[2] I. Kaplansky, Maximal fields with valuations I and II, Duke Math. J. (Vol. 9, 1942, 303–321) and

(Vol. 12, 1945 243–248).

[3] S. Kuhlmann & M. Serra, The automorphism group of a valued field of generalised formal power

series, J. Algebra, Vol. 605, 2022, Pages 339–376, DOI:10.1016/j.jalgebra.2022.04.023.

[4] O. F. G. Schilling, Automorphisms of fields of formal power series, Bull. Amer. Math. Soc. 50 (12)

(1944) 892–901.

[5] M. Serra, Automorphism groups of hahn groups and hahn fields, Ph.D. thesis, Universit¨at Konstanz (2021).

http://nbn-resolving.de/urn:nbn:de:bsz:352-2-19yqyhucnbtlw