Abstract
We consider the completion of a topological field whose topology is defined by the valuations of a dependence class of valuation rings. We characterize the separable closure of the field in its completion through henselization of those valuation rings. Actually, we prove that this relative separable closure is the fixed field of some particular subgroup of continuous automorphisms of the total separable closure. We prove also that this relative separable closure has some properties which are analogous to those of the henselization.
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Literature
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This paper is part of the author's doctoral dissertation and was finished during his stay in Konstanz though the CNPq/GMD program.
The results 2.11 and 2.12 were firstly announced in C.R. Acad. Sc. Paris, t.283 (1977).
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Engler, A.J. The relative separable closure of a valued field in its completion. Manuscripta Math 24, 83–95 (1978). https://doi.org/10.1007/BF01168564
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DOI: https://doi.org/10.1007/BF01168564