Immediate and purely wild extensions of valued fields

Kuhlmann, Franz-Viktor; Pank, Matthias; Roquette, Peter

doi:10.1007/bf01168612

Cited by 50 publications

(42 citation statements)

References 12 publications

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“…An algebraic extension F 0 ≥ F is tame (see [3,4]) if, for every intermediate subexten- For finite extensions, this follows directly from the relations…”

Section: Hensel Lemma Let F ∈ R[x] Be a Monic Polynomial Such That Tmentioning

confidence: 99%

“…An extension F 0 ≥ F is said to be purely wild (see [3,4]) if F R 0 is a purely inseparable extension of F R and the quotient group Γ R 0 /Γ R is a p-group. Remark 1.…”

Section: Corollary 2 (To the Definition) If F Is Algebraically Complmentioning

confidence: 99%

“…The following theorem of Pank (see [3]) on splitting of the algebraic closure is of great importance.…”

Section: Theorem 1 For Every Algebraic Extensionmentioning

confidence: 99%

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Tame and purely wild extensions of valued fields

Ershov

2008

St. Petersburg Math. J.

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Abstract. A systematic and concise exposition of the basic results concerning two complementary classes (tame and purely wild) of extensions of (Henselian) valued fields is given. These notions proved to be quite useful both for the general theory and for the model theory of such fields. Along with new results, new proofs of old results are presented. Thus, in the proof of the well-known Pank theorem on the existence of a complement to the ramification group in the absolute Galois group of a Henselian valued field, the properties of maximal immediate extensions are employed instead of cohomological methods.

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“…An algebraic extension F 0 ≥ F is tame (see [3,4]) if, for every intermediate subexten- For finite extensions, this follows directly from the relations…”

Section: Hensel Lemma Let F ∈ R[x] Be a Monic Polynomial Such That Tmentioning

confidence: 99%

“…An extension F 0 ≥ F is said to be purely wild (see [3,4]) if F R 0 is a purely inseparable extension of F R and the quotient group Γ R 0 /Γ R is a p-group. Remark 1.…”

Section: Corollary 2 (To the Definition) If F Is Algebraically Complmentioning

confidence: 99%

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Tame and purely wild extensions of valued fields

Ershov

2008

St. Petersburg Math. J.

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“…It was established by Kuhlmann, Pank, and Roquette, who generalized a result of Kaplansky. (See [9], Theorem 5.1. One cannot do without the assumption on the residue field.)…”

mentioning

confidence: 99%

Quantifier elimination in tame infinite p-adic fields

Brigandt

2001

J. symb. log.

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Abstract. We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp ('infinite p-adic fields') using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky's condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued fields is extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient. §1. Introduction. When the model theory of p-adic fields started around 1965, the analogy of Q p with the real numbers played an important role. 1 Indeed, James Ax and Simon Kochen [1], as well as independently Yuri Ershov [3] established the decidability of the theory of p-adic numbers. Ax and Kochen set up a complete axiom system for Q p in the language of valued fields and showed that it is model complete. However, the theory of Q p does not admit quantifier elimination in the language of valued fields-whereas the theory of real closed fields admits elimination of quantifiers in the language of ordered fields. This is due to the fact that two p-adic closures of a given formally p-adic field are in general not isomorphic over the latter. For this reason, it is necessary to extend the language of valued fields in order to obtain quantifier elimination.The pioneering work of Ax and Kochen included a quantifier elimination result for the theory of Q p (see [2]). They extended the language by countably many functions f n and a cross-section π. 2 In 1976, Angus Macintyre [10] was able to give an improved result by finding a language with which it is more straightforward to deal. He enlarged the language of valued fields by power predicates P n in order to obtain elimination of quantifiers (P n denotes the subset of n-th powers, where n ∈ N). Seven years later the finding of Macintyre was generalized by Alexander Prestel and Peter Roquette [12] to finite extensions of Q p . They made use of P n predicates and additional constants.

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“…It is because of the latter fact that the uniqueness of maximal immediate extensions will in general fail (cf. [12]). This is what makes the proof of the model theoretic results for tame and separably tame fields much harder than for algebraically and separable-algebraically maximal Kaplansky fields.…”

Section: Introductionmentioning

confidence: 99%

The model theory of separably tame valued fields

Kuhlmann

Pal

2016

Journal of Algebra

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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.

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Immediate and purely wild extensions of valued fields

Cited by 50 publications

References 12 publications

Tame and purely wild extensions of valued fields

Tame and purely wild extensions of valued fields

Quantifier elimination in tame infinite p-adic fields

The model theory of separably tame valued fields

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