Abstract
We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author’s proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.
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Acknowledgments
The authors wish to thank the referee for his thorough reading of the paper and several very helpful suggestions. The first author wishes to thank Peter Roquette for his invaluable help and support during the preparation of his doctoral thesis, in which several of the results presented here appeared first. He also wishes to thank F. Delon, B. Green, A. Prestel and F. Pop for inspiring discussions and suggestions. Thanks also to Salih Durhan for his careful reading of the manuscript. During the work on this paper, the first author was partially supported by a Canadian NSERC Grant
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Kuhlmann, FV., Vlahu, I. The relative approximation degree in valued function fields. Math. Z. 276, 203–235 (2014). https://doi.org/10.1007/s00209-013-1194-1
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DOI: https://doi.org/10.1007/s00209-013-1194-1