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The algebra and model theory of tame valued fields
- Source :
- Journal für die reine und angewandte Mathematik (Crelles Journal). 2016:1-43
- Publication Year :
- 2014
- Publisher :
- Walter de Gruyter GmbH, 2014.
-
Abstract
- A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be 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. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.
- Subjects :
- Algebraic function field
Applied Mathematics
General Mathematics
010102 general mathematics
Normal extension
Field (mathematics)
Commutative Algebra (math.AC)
Mathematics - Commutative Algebra
01 natural sciences
Algebraic closure
Algebra
Residue field
Algebraic theory
0103 physical sciences
FOS: Mathematics
12J10, 12J15
010307 mathematical physics
0101 mathematics
Algebraically closed field
Valuation (algebra)
Mathematics
Subjects
Details
- ISSN :
- 14355345, 00754102, and 20140029
- Volume :
- 2016
- Database :
- OpenAIRE
- Journal :
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Accession number :
- edsair.doi.dedup.....28f0ac260bee15b712bed67f6655ce5d