The model theory of separably tame valued fields

Abstract

Abstract. A henselian valued field K is called separably tame if its separable-algebraic closure K sep is a tame extension, that is, the ramification field of the normal extension K sep |K is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove AxKochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.

Elimination of ramification II: Henselian rationality

“…Proof. It is shown in the proof of [20,Proposition 3.10] that under the assumptions of our lemma, the following holds: every pseudo Cauchy sequence in (K, v) with limit x and without a limit in K is of transcendental type; for these notions, see [9]. This implies (and in fact is equivalent to) that the approximation type of x over K is transcendental.…”

“…In the same paper and in [20], Theorems 1.2 and 1.3 are also used to prove other Ax-Kochen-Ershov Principles (which then also hold in mixed characteristic), and further model theoretic results for tame and separably tame valued fields. The reader should note that in the present paper we will make extensive use of the valuation theoretical preliminaries and the general algebraic theory of tame and separably tame fields presented in Sections 2 and 3 of [17].…”

Infinite towers of Artin-Schreier defect extensions of rational function fields

“…, and [Ku5]). In Section 4.1 we prove the following fact which is a generalization of Theorem 1.2 of [Ku1].…”

Characterizing NIP henselian fields

“…For the case of infinite degree of imperfection, note that by [10, Théorème 3.1], the theory of separably algebraically maximal Kaplansky valued fields of characteristic $$p$$ and given imperfection degree (allowed to be infinite) with value group elementarily equivalent to $$vK$$ and residue field elementarily equivalent to $$Kv$$ is complete. Exactly as explained in the proof of [22, Lemma 3.1], the theory of $$(K,v)$$ satisfies (SE) also in case the degree of imperfection is infinite; only citing Delon ([10, Théorème 3.1]) rather than Kuhlmann and Pal ([29, Theorem 5.1]). In [22, Lemma 3.2], (Im) is proved in the case that the imperfection degree of $$K$$ is finite.…”