Submission date: 10 September 2014.

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 Ax-Kochen-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.

Mathematics Subject Classification: Primary 03C10, 12J10, Secondary 03C60, 12J20

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Full text arXiv 1409.3141: pdf, ps.