Advertisement

Distances of elements in valued field extensions

  • Anna BlaszczokEmail author
Open Access
Article
  • 29 Downloads

Abstract

We develop a modification of a notion of distance of an element in a valued field extension introduced by F.-V. Kuhlmann. We show that the new notion preserves the main properties of the distance and at the same time gives more complete information about a valued field extension. We study valued field extensions of prime degree to show the relation between the distances of the elements and the corresponding extensions of value groups and residue fields. In connection with questions related to defect extensions of valued function fields of positive characteristic, we present constructions of defect extensions of rational function fields K(xy)|K generated by elements of various distances from K(xy). In particular, we construct dependent Artin–Schreier defect extensions of K(xy) of various distances.

Mathematics Subject Classification

12J10 12J25 

Notes

References

  1. 1.
    Blaszczok, A.: On the structure of immediate extensions of valued fields, Ph.D. thesis, University of Silesia (2014)Google Scholar
  2. 2.
    Blaszczok, A.: Infinite towers of Artin–Schreier defect extensions of rational function fields. In: Second International Conference and Workshop on Valuation Theory (Segovia/El Escorial, Spain, 2011), EMS Series of Congress Reports, vol. 10, pp. 16–54 (2014)Google Scholar
  3. 3.
    Blaszczok, A., Kuhlmann, F.-V.: Algebraic independence of elements in immediate extensions of valued fields. J. Algebra 425, 179–214 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blaszczok, A., Kuhlmann, F.-V.: Corrections and notes to “Value groups, residue fields and bad places of rational function fields”. Trans. Am. Math. Soc. 3675, 4505–4515 (2015)CrossRefGoogle Scholar
  5. 5.
    Blaszczok, A., Kuhlmann, F.-V.: On maximal immediate extensions of valued fields. Math. Nachr. 290, 7–18 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blaszczok, A., Kuhlmann, F.-V.: Counting of the number of distinct distances of elements in valued field extensions. J. Algebra 509, 192–211 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cutkosky, S.D., Piltant, O.: Ramifcation of valuations. Adv. Math. 183, 1–79 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Endler, O.: Valuation Theory. Springer, Berlin (1972)CrossRefGoogle Scholar
  9. 9.
    Engler, A.J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  10. 10.
    Hahn, H.: Über die nichtarchimedischen Größensysteme. S.-B. Akad. Wiss. Wien Math.-naturw. Kl. Abt. IIa 116, 601–655 (1907)zbMATHGoogle Scholar
  11. 11.
    Kaplansky, I.: Maximal fields with valuations I. Duke Math. J. 9, 303–321 (1942)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Karpilovsky, G.: Topics in Field Theory. Mathematics Studies, vol. 155. North Holland, Amsterdam (1989)Google Scholar
  13. 13.
    Khanduja, S.K., Singh, A.P.: On theorem of Tignol for defectless extensions and its converse. J. Algebra 288, 400–408 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krull, W.: Allgemeine bewertungstheorie. J. Reine Angew. Math. 167, 160–196 (1932)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kuhlmann, F.-V.: Value groups, residue fields and bad places of rational function fields. Trans. Am. Math. Soc. 356, 4559–4600 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kuhlmann, F.-V.: Additive polynomials and their role in the model theory of valued fields. In: Logic in Tehran, Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, held October 18–22, 2003. Lecture Notes in Logic, vol. 26, pp. 160–203 (2006)Google Scholar
  17. 17.
    Kuhlmann, F.-V.: A classification of Artin–Schreier defect extensions and a characterization of defectless fields. Ill. J. Math. 54, 397–448 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kuhlmann, F.-V.: Approximation of elements in henselizations. Manuscr. Math. 136, 461–474 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kuhlmann, F.-V.: Defect. In: Fontana, M., Kabbaj, S.-E., Olberding, B., Swanson, I. (eds.) Commutative Algebra–Noetherian and non-Noetherian Perspectives, pp. 277–318. Springer, Berlin (2011)Google Scholar
  20. 20.
    Kuhlmann, F.-V.: The model theory of tame valued fields. J. Reine Angew. Math. 719, 1–43 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kuhlmann, F.–V.: Valuation Theory, Book in Preparation. http://math.usask.ca/~fvk/Fvkbook.htm
  22. 22.
    Kuhlmann, F.-V. and Piltant, O., Higher ramification groups for Artin–Schreier defect extensions. (in preparation)Google Scholar
  23. 23.
    Kuhlmann, F.-V., Vlahu, I.: The relative approximation degree in valued function fields. Math. Z. 276, 203–235 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lang, S.: Algebra Graduate Texts in Mathematics. Springer, New York (2002)Google Scholar
  25. 25.
    MacLane, S., Schilling, O.F.G.: Zero-dimensional branches of rank 1 on algebraic varieties. Ann. Math. 40, 507–520 (1939)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Warner, S.: Topological Fields. Mathematics Studies, vol. 157. North Holland, Amsterdam (1989)Google Scholar
  27. 27.
    Zariski, O., Samuel, P.: Commutative Algebra. vol. II. New York (1960)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of SilesiaKatowicePoland

Personalised recommendations