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Preprint Number 587

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587. Antongiulio Fornasiero, Franz-Viktor Kuhlmann, Salma Kuhlmann
Towers of complements to valuation rings and truncation closed embeddings of valued fields
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Submission date: 27 May 2013.

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 ({\it tower of complements}) of k-subspaces of K which are complements of the (possibly fractional) ideals of the valuation ring. 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.

Mathematics Subject Classification: 12J10, 12J15, 12L12, 13A18, Secondary: 03C60, 12F05, 12F10, 12F20
Journal-ref: J. Algebra 323 (2010), no. 3, 574-600

Keywords and phrases:

Full text arXiv 1305.6304: pdf, ps.


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