Fields with automorphism and valuation
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Mathematical Logic
Fields with automorphism and valuation Özlem Beyarslan1 · Daniel Max Hoffmann2,3 · Gönenç Onay4 · David Pierce5 Received: 19 December 2018 / Accepted: 18 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The model companion of the theory of fields with valuation and automorphism (of the pure field structure) exists. A counterexample shows that the theory of models of ACFA equipped with valuation is not this model companion. Keywords Difference fields · Valued fields · Model companion Mathematics Subject Classification Primary 03C60; Secondary 12J10 · 12H10
1 Introduction The question has been of interest for decades: If a model-complete L-theory T is augmented with axioms for an automorphism σ of its models, does the resulting L σ theory Tσ have a model companion? It does have, when T is the theory ACF of algebraically closed fields. The model companion of Tσ in this case is ACFA, studied by Macintyre [1] and Chatzidakis and Hrushovski [2] and others. However, Tσ is not companionable when T is ACFA itself [3]. A result in general model theory, established by Kikyo [4], is that if Tσ is companionable, and T is dependent, then T must also be stable. In particular then, Tσ cannot be companionable when T is the theory ACVF of algebraically closed valued fields in the signature of fields with a predicate for a valuation ring (inducing a non-trivial valuation).
Daniel Max Hoffmann: SDG. Supported by the Polish Natonal Agency for Academic Exchange and the National Science Centre (Narodowe Centrum Nauki, Poland) grant nos. 2016/21/N/ST1/01465, and 2015/19/B/ST1/01150.
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David Pierce [email protected] http://mat.msgsu.edu.tr/∼dpierce/
Extended author information available on the last page of the article
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In some cases where T is a model-complete theory of valued fields, Tσ may become companionable when an additional condition is imposed on σ . Note that an automorphism σ of a valued field induces an automorphism σv of the the value group and an automorphism σ¯ of the residue field. When T is the model companion of the theory of any of the following classes of valued fields, then Tσ becomes companionable under the given condition on σ : (1) isometric valued difference fields, where σv (γ ) = γ for all γ in [5,6], (2) contractive valued difference fields, where σv (γ ) > nγ for all positive γ in and n in ω [7], (3) multiplicative valued fields, where σv (γ ) = ργ for all γ in , for a certain constant ρ [8]. Moreover, (4) Tσ ∪ Tv is companionable when T is ACVF and Tv is a companionable theory of ordered abelian groups equipped with an automorphism [9] (in this case (, σv ) | Tv ). The corresponding model companion in each of the four cases satisfies an analogue of Hensel’s lemma for σ -polynomials (see [7, Definition 4.2]). Treating automorphisms and derivations as instances of an operator called D, Scanlon [10] establishes a model-complete theory of fields equipped with D and a valuation having certain interactions. In his doctora
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