Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-22T14:30:11.266Z Has data issue: false hasContentIssue false
  • English
  • Français

A GALOIS THEORY FOR THE FIELD EXTENSION K((X))/K

Published online by Cambridge University Press:  25 August 2010

ANGEL POPESCU ,
ASIM NASEEM  and
NICOLAE POPESCU
ANGEL POPESCU
Affiliation:
Technical University of Civil Engineering Bucharest, Department of Mathematics and Computer Science, B-ul Lacul Tei 124, 020396 Bucharest 38, Romania e-mail: angel.popescu@gmail.com
ASIM NASEEM
Affiliation:
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan e-mail: asimroz@gmail.com
NICOLAE POPESCU
Affiliation:
Mathematical Institute of the Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania e-mail: Nicolae.Popescu@imar.ro
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.

Keywords

Primary 12F1013F2512J20Secondary 12J1012E9912F99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 447 - 451
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Artin, E., Algebraic Numbers and Algebraic Functions (Gordon and Breach, Sciences Publishers, New York, 1967).Google Scholar
2.Dummit, D. S. and Foote, R. M., Abstract Algebra, 2nd edn. (John Wiley & Sons, Inc. New York, 2003).Google Scholar
3.Neukirch, J., Algebraic Number Theory (Springer Verlag, Berlin, 1999).CrossRefGoogle Scholar
4.Roquette, P., On the History of Valuation Theory, Part I in Valuation Theory and its Applications, Vol. I (Kuhlmann, F. V., Kuhlmann, S. and Marshal, M., Editors), (Fields Institute Communications, AMS, New York, 2002), 291–355.Google Scholar
5.Schilling, O. F. G., Automorphisms of fields of formal power series, Bull. Amer. Math. Soc. 50 (12) (1944), 892–901.CrossRefGoogle Scholar
6.Schmidt, F. K., Mehrfach perfekte Körper, Math. Ann. 108 (1933), 1–25.CrossRefGoogle Scholar
7.Weiss, E., Algebraic Number Theory (McGraw-Hill Book Company, Inc., New York, 1963).Google Scholar