Infinite Towers of Galois Defect Extensions of Kaplansky Fields

Anna Blaszczok

doi:10.1515/amsil-2017-0012

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Infinite Towers of Galois Defect Extensions of Kaplansky Fields

Annales Mathematicae Silesianae

Volume 32 (2018): Issue 1 (September 2018)

By: Anna Blaszczok

Open Access

|Aug 2018

Abstract

We give conditions for Kaplansky fields to admit infinite towers of Galois defect extensions of prime degree. As proofs of the presented facts are constructive, this provides examples of constructions of infinite towers of Galois defect extensions of prime degree. We also give a constructive proof of the fact that a henselian Kaplansky field cannot be defectless-by-finite.

References

[1] Blaszczok A., Infinite towers of Artin-Schreier defect extensions of rational function fields, in: A. Campillo, F.-V. Kuhlmann, B. Teissier (Eds.), Valuation theory in interaction, EMS Series of Congress Reports, European Mathematical Society (EMS), Zürich, 2014, pp. 16-54.10.4171/149-1/3
Search in Google Scholar Back to article
[2] Blaszczok A., Distances of elements in valued field extensions, submitted.
Search in Google Scholar Back to article
[3] Blaszczok, A., Kuhlmann F.-V., Algebraic independence of elements in immediate extensions of valued fields, J. Algebra 425 (2015), 179-214.10.1016/j.jalgebra.2014.10.050
Search in Google Scholar Back to article
[4] Blaszczok A., Kuhlmann F.-V., On maximal immediate extensions of valued fields, Math. Nachr. 290 (2017), 7-18.10.1002/mana.201500073
Search in Google Scholar Back to article
[5] Endler O., Valuation theory, Springer-Verlag, Berlin, 1972.10.1007/978-3-642-65505-0
Search in Google Scholar Back to article
[6] Kaplansky I., Maximal fields with valuations, Duke Math. J. 9 (1942), 303-321.
Search in Google Scholar Back to article
[7] Karpilovsky G., Topics in field theory, North-Holland Mathematics Studies 155, North-Holland Publishing Co., Amsterdam, 1989.
Search in Google Scholar Back to article
[8] Krull W., Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196.
Search in Google Scholar Back to article
[9] Kuhlmann F.-V., Henselian function fields and tame fields, preprint (extended version of Ph.D. thesis), Heidelberg, 1990.
Search in Google Scholar Back to article
[10] Kuhlmann F.-V., Valuation theoretic and model theoretic aspects of local uniformization, in: H. Hauser, J. Lipman, F. Oort, A. Quirós (Eds.), Resolution of singularities. A research textbook in tribute to Oscar Zariski, Progress in Mathematics 181, Birkhäuser Verlag, Basel, 2000, pp. 381-456.10.1007/978-3-0348-8399-3_15
Search in Google Scholar Back to article
[11] Kuhlmann F.-V., Value groups, residue fields, and bad places of rational function fields, Trans. Amer. Math. Soc. 356 (2004), 4559-4600.10.1090/S0002-9947-04-03463-4
Search in Google Scholar Back to article
[12] Kuhlmann F.-V., A classification of Artin-Schreier defect extensions and characterizations of defectless fields, Illinois J. Math. 54 (2010), 397-448.
Search in Google Scholar Back to article
[13] Kuhlmann F.-V., The defect, in: M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson (Eds.), Commutative algebra. Noetherian and non-Noetherian perspectives, Springer- Verlag, New York, 2011, pp. 277-318.10.1007/978-1-4419-6990-3_11
Open DOI Search in Google Scholar Back to article
[14] Kuhlmann F.-V., The algebra and model theory of tame valued fields, J. Reine Angew. Math. 719 (2016), 1-43.
Search in Google Scholar Back to article
[15] Kuhlmann F.-V., Elimination of ramification II: henselian rationality of valued function fields, in preparation.
Search in Google Scholar Back to article
[16] Whaples G., Galois cohomology of additive polynomial and n-th power mappings of fields, Duke Math. J. 24 (1957), 143-150.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/amsil-2017-0012 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

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Language: English

Page range: 65 - 78

Submitted on: Apr 1, 2017

Accepted on: Nov 8, 2017

Published on: Aug 24, 2018

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

defect extensions,

Kaplansky fields,

Artin-Schreier defect extensions,

Kummer defect extensions,

defectless-by-finite fields

Related subjects:

Mathematics,

General mathematics

© 2018 Anna Blaszczok, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 32 (2018): Issue 1 (September 2018)