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The algebra and model theory of tame valued fields. (English) Zbl 1401.13011

The purpose of this article is to extend to new classes of valued fields results similar to those of Ax-Kochen-Ershov, which prove that the elementary theory of a valued field \((K,v)\) is determined by the theories of its valued group \(vK\) and of its residue field \(Kv\). In general, the proofs of these theorems use the uniqueness of the maximal immediate algebraic extension of certain valued fields (i.e. \(L|K\) is algebraic, \(vL=vK\) and \(Lv=Kv\)). What is new here is that in the case of tame fields this maximal immediate algebraic extension is not necessarily unique.
The author defines three AKE principles. The first one is called \(\mathrm{AKE}^\equiv\) Principle: \[ vK\equiv vL\wedge Kv\equiv Lv \Rightarrow (K,v)\equiv (L,v), \] where \(\equiv\) denotes elementary extension. By replacing elementary equivalence by elementary inclusion (\(\prec\)), one obtains \(\mathrm{AKE}^\prec\) Principle: \[ (K,v)\subset (L,v)\wedge vK\prec vL\wedge Kv\prec Lv \Rightarrow (K,v)\prec (L,v). \] Finally, by replacing \(\prec\) by \(\prec_\exists\) in the previous formula one gets \(\mathrm{AKE}^\exists\) Principle, where \(K\prec_\exists L\) holds if every existential formula with parameters from \(K\) which holds in \(L\) also holds in \(K\).
Before quoting the main results we recall some definitions. A valued field is henselian if it admits a unique extension of the valuation to every algebraic extension field; a classical result proves that this is a first-order property. If \(L\) is a finite algebraic extension of a henselian valued field, then \(L|K\) is said to be defectless if \([L:K]=(vL:vK)[Lv:Kv]\). The henselian valued field \((K,v)\) is called defectless if each of its finite extensions is defectless, and separably defectless if each of its finite separable extension is defectless. The author says that the extension \((L|K,v)\) (where \(L|K\) has finite transcendence degree) is without transcendence defect if equality holds in Abhyankar’s inequality: \[ \mathrm{trdeg}\; L|K\geq \mathrm{trdeg}\;Lv|Kv+\mathrm{dim}_{\mathbb{Q}} \mathbb{Q}\otimes vL/vK. \] Now, the author proves that every extension without transcendence defect of a henselian defectless field satisfies \(\mathrm{AKE}^\exists\) Principle. The henselian valued field \((K,v)\) is tame if it is defectles, and for every finite algebraic extension \(L|K\) the characteristic exponent of \(Kv\) is prime to \((vL:vK)\) and \(Lv| Kv\) is a separable extension. A separably tame field is a separably defectless field such that for every finite separable algebraic extension \(L|K\) the characteristic exponent of \(Kv\) is prime to \((vL:vK)\) and \(Lv| Kv\) is a separable extension. The class of tame fields is an elementary class. The author proves that the tame fields satisfy \(\mathrm{AKE}^\exists\) Principle and \(\mathrm{AKE}^\prec\) Principle; furthermore, the tame fields of equal characteristic satisfy \(\mathrm{AKE}^\equiv\) Principle (this theorem has already been proved in his PhD). Finally, he proves the following two results. Every separable extension \((L|K,v)\) without transcendence defect of a henselian separably defectless field, such that \(vK\) is cofinal in \(vL\), satisfies \(\mathrm{AKE}^\exists\) Principle. Every separable extension of a separably tame field satisfies \(\mathrm{AKE}^\exists\) Principle.

MSC:

13A18 Valuations and their generalizations for commutative rings
12J10 Valued fields
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