Towers of complements to valuation rings and truncation closed embeddings of valued fields

Fornasiero, Antongiulio; Kuhlmann, Franz-Viktor; Kuhlmann, Salma

Towers of complements to valuation rings and truncation closed embeddings of valued fields

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Date

2010

Authors

Fornasiero, Antongiulio

Kuhlmann, Franz-Viktor

Kuhlmann, Salma

Publication type

Journal article

Published in

Journal of Algebra ; 323 (2010), 3. - pp. 574-600. - ISSN 0021-8693

Abstract

We study necessary and sufficient conditions for a valued field K with value group G and residue field k (with char K=char k) to admit a truncation closed embedding in the field of generalized power series k((G,f)) (with factor set f). We show that this is equivalent to the existence of a family (tower of complements) of k-subspaces of K which are complements of the (possibly fractional) ideals of the valuation ring, and satisfying certain natural conditions. If K is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.

Subject (DDC)

510 Mathematics

Keywords

Completion of an ordered group,Valued field,Fields of generalized power series,Truncation closed embedding,Complement to valuation ring

Cite This

ISO 690

FORNASIERO, Antongiulio, Franz-Viktor KUHLMANN, Salma KUHLMANN, 2010. Towers of complements to valuation rings and truncation closed embeddings of valued fields. In: Journal of Algebra. 323(3), pp. 574-600. ISSN 0021-8693. Available under: doi: 10.1016/j.jalgebra.2009.11.023

BibTex

@article{Fornasiero2010Tower-12753,
  year={2010},
  doi={10.1016/j.jalgebra.2009.11.023},
  title={Towers of complements to valuation rings and truncation closed embeddings of valued fields},
  number={3},
  volume={323},
  issn={0021-8693},
  journal={Journal of Algebra},
  pages={574--600},
  author={Fornasiero, Antongiulio and Kuhlmann, Franz-Viktor and Kuhlmann, Salma}
}

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Yes