Automorphisms of fields of formal power series

Schilling, O. F. G.

doi:10.1090/s0002-9904-1944-08259-1

Cited by 9 publications

(10 citation statements)

References 9 publications

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“…Theorem 4.2.8 thus provides a decomposition of v-Aut + (k) K and v-Aut + k K purely in terms of the valuation invariants of K. In the next section we are going to apply the results obtained so far under some further assumptions on the group G and the field k. This will allow to retrieve results of Schilling [Sch44] on the field of Laurent series and of Deschamps [Des05] on the field of Puiseux series.…”

Section: Finally We Havementioning

confidence: 96%

“…Inspired by the work of Schilling [Sch44], in this section we study automorphisms of a Hahn field K which have the very powerful property of commuting with infinite sums.…”

Section: Strongly Additive Automorphismsmentioning

confidence: 99%

“…Pursuing our generalisation of Schilling's results, we study the group v-Aut + K of strongly additive (Definition 4.0.1) valuation preserving automorphisms (and its subgroups v-Aut + (k) K, v-Aut + k K) of a Hahn field K. In particular, in Theorem 4.2.8 we describe v-Aut + (k) K, for a Hahn field K which satisfies the first and second lifting property, in terms of the valuation invariants of K. Finally, to illustrate our results, we provide a detailed study of two special cases. If the group G is finitely generated we obtain a generalisation of a result of [Sch44] on the field L of Laurent series. If G is divisible and finite dimensional, we obtain increasingly precise results depending on our assumptions on k. For the field P of Puiseux series we find descriptions of v-Aut (k) P and v-Aut k P that generalise results of [Des05].…”

Section: Introductionmentioning

confidence: 95%

“…In his paper [Sch44], Schilling studied the k-automorphisms of the field L = k((Z)) of formal Laurent series over a coefficient field k. First, he observed [Sch44, Lemma 1] that all k-automorphisms are necessarily valuation preserving. Moreover, since the ordered abelian group Z admits only the trivial automorphism, all k-automorphisms of L are internal (Definition 3.1.1).…”

Section: Introductionmentioning

confidence: 99%

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

Kuhlmann,

Serra

2021

Preprint

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Let k be a field, G a totally ordered abelian group and K = k((G)) the maximal field of generalised power series, endowed with the canonical valuation v [Hah07]. 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 are able to generalise and refine the decomposition of v-Aut K [Hof91] and prove a structure theorem decomposing v-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 Hom(G, k × ) of G into k × and 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 [Sch44] on the field of Laurent series and that of [Des05] on the field of Puiseux series.

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Section: Finally We Havementioning

confidence: 96%

“…Inspired by the work of Schilling [Sch44], in this section we study automorphisms of a Hahn field K which have the very powerful property of commuting with infinite sums.…”

Section: Strongly Additive Automorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kuhlmann,

Serra

2021

Preprint

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“…Throughout the paper, Hahn groups and Hahn fields play a fundamental role. The group of automorphisms of Hahn structures have been extensively studied, see [3], [8], [11] and [26]. In future work, we will analyse the behaviour of the difference rank as function defined on these automorphism groups.…”

Section: Introductionmentioning

confidence: 99%

The Valuation Difference Rank of a Quasi-Ordered Difference Field

Kuhlmann¹,

Matusinski²,

Point³

2017

Groups, Modules, and Model Theory - Surveys and Recent Developments

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There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued difference field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of difference rank. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of a maximally valued quasi-ordered difference field. As an application, we show that for every regular uncountable cardinal κ such that κ = κ <κ , there are 2 κ pairwise non-isomorphic quasi-ordered difference fields of cardinality κ, but all isomorphic as quasi-ordered fields.

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Normal subgroups of the one-dimensional group of formal-analytic transformations over a local ring

Shramko

1979

Ukr Math J

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Automorphisms of fields of formal power series

Cited by 9 publications

References 9 publications

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

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

The Valuation Difference Rank of a Quasi-Ordered Difference Field

Normal subgroups of the one-dimensional group of formal-analytic transformations over a local ring

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