Fields with two incomparable henselian valuation rings

Engler, Antonio José

doi:10.1007/bf01167696

Cited by 12 publications

(3 citation statements)

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“…This result follows from a theorem of Schimdt, that was generalized by Engler, and later was reproved by Jarden, see [8].…”

Section: The Galois Closure Of Pac Extensionssupporting

confidence: 50%

On pseudo algebraically closed extensions of fields

Bary-Soroker

2009

Journal of Algebra

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The notion of 'Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. As a result we get a classification of all finite PAC extensions which in turn proves the 'bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g. transitiveness of PAC extensions.

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“…This result follows from a theorem of Schimdt, that was generalized by Engler, and later was reproved by Jarden, see [8].…”

Section: The Galois Closure Of Pac Extensionssupporting

confidence: 50%

On pseudo algebraically closed extensions of fields

Bary-Soroker

2009

Journal of Algebra

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“…Proof. This is well-known; see, for example, the first section of [4]. A proof can be quite easily assembled from various results in [6, Section 5].…”

Section: Note Thatmentioning

confidence: 78%

On logical characterization of henselianity

Yin

2007

Preprint

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We give some sufficient conditions under which any valued field that admits quantifier elimination in the Macintyre language is henselian. Then, without extra assumptions, we prove that if a valued field of characteristic (0, 0) has a Z-group as its value group and admits quantifier elimination in the main sort of the Denef-Pas style language LRRP then it is henselian. In fact the proof of this suggests that a quite large class of Denef-Pas style languages is natural with respect to henselianity. §1. Introduction. One of the most important tools in model-theoretic algebra is quantifier elimination (QE). Tarski's Theorem laid the foundation for the subsequent work along this line:Theorem 1.1 (Tarski). The theory RCF of real closed fields, as formulated in the language L OR of ordered rings, admits QE.Much later Macintyre proved a very important analog of this result for p-adic fields in [9]: Theorem 1.2 (Macintyre). The theory of p-adic fields, as formulated in the language L Mac , admits QE.A crucial question for the algebraic structure of a field is of course under what conditions polynomials have roots. Properties that answer this question in real closed fields and p-adic fields are essential to the proofs of the above two theorems. They are of course real-closedness and henselianity, respectively. One may raise the question: Is QE equivalent to these properties after all? For real closed fields there is a good answer: Theorem 1.3 (Macintyre, McKenna, van den Dries). Let K be an ordered field such that the theory of K in L OR admits QE. Then K is real closed.This result is established in [10], in which the authors actually give a quite general technique that can be used to establish other similar "converse QE" results for various kinds of fields. In particular they have the following analogous result for p-fields:

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“…If a field K has a henselian valuation v and L is a subfield of K with [K : L] < ∞, then the restriction w = v| L need not be henselian. But it is easy to see that w is then 'semihenselian,' that is, w has more than one but only finitely many different extensions to a separable closure L sep of L. See [2] for a thorough analysis of semihenselian valuations. Notably, Engler shows that w is semihenselian if and only if the residue field L w is algebraically closed but there is a henselian valuation u on L such that u is a proper coarsening of w and the residue field L u is real closed.…”

Section: Henselian To Graded Reductionmentioning

confidence: 99%

Unitary SK₁ of graded and valued division algebras

Hazrat

Wadsworth

2011

Proceedings of the London Mathematical Society

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Abstract. The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.

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Fields with two incomparable henselian valuation rings

Cited by 12 publications

References 10 publications

On pseudo algebraically closed extensions of fields

On pseudo algebraically closed extensions of fields

On logical characterization of henselianity

Unitary SK₁ of graded and valued division algebras

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Fields with two incomparable henselian valuation rings

Cited by 12 publications

References 10 publications

On pseudo algebraically closed extensions of fields

On pseudo algebraically closed extensions of fields

On logical characterization of henselianity

Unitary SK1 of graded and valued division algebras

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Resources

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Unitary SK₁ of graded and valued division algebras