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Israel Journal of Mathematics

, Volume 198, Issue 1, pp 229–242 | Cite as

Embedding problems with local conditions and the admissibility of finite groups

  • NguyỄn Duy Tân
Article
  • 135 Downloads

Abstract

Let K be a field of characteristic p > 0, which has infinitely many discrete valuations. We show that every finite embedding problem for Gal(K) with finitely many prescribed local conditions, whose kernel is a p-group, is properly solvable. We then apply this result in studying the admissibility of finite groups over global fields of positive characteristic. We also give another proof for a result of Sonn.

Keywords

Weak Solution Exact Sequence Galois Group Division Algebra Sylow Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BT]
    L. Bary-Soroker and N. D. Tan, On p-embedding problems in characteristic p, Journal of Pure and Applied Algebra 215 (2011), 2533–2537.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [CF]
    J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, London, 1967.zbMATHGoogle Scholar
  3. [CGP]
    B. Conrad, O. Gabber and G. Prasad, Pseudo-reductive groups, Series: New Mathematical Monographs (No. 17), Cambridge University Press, Cambridge, 2010.Google Scholar
  4. [Du]
    S. Durhan, Additive polynomials over perfect fields, available at: http://math.ncc.metu.edu.tr/content/files/azgin/addpol.pdf
  5. [Ef]
    I. Efrat, Valuations, Orderings, and Milnor K-theory, Mathematical Surveys and Monographs 124, American Mathematical Society, Providence, RI, 2006.Google Scholar
  6. [FS]
    B. Fein and M. Schacher, Galois groups and division algebras, Journal of Algebra 38 (1976), 182–191.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [FrJ]
    M. D. Fried and M. Jarden, Field Arithmetic, Third edition, revised by Moshe Jarden, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] (3) 11, Springer-Verlag, Berlin, 2008.zbMATHGoogle Scholar
  8. [Ha1]
    D. Harbater, Embedding problems with local conditions, Israel Journal of Mathematics 118 (2000), 317–355.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Ha2]
    D. Harbater, Correction and addendum to “Embedding problems with local conditions”, Israel Journal of Mathematics 162 (2007), 373–379.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [HHK]
    D. Harbater, J. Hartmann and D. Krashen, Patching subfields of division algebras, Transactions of the American Mathematical Society 363 (2011), 3335–3349.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Ja]
    M. Jarden, On p-embedding problems in characteristic p, private note.Google Scholar
  12. [La]
    S. Lang, Algebra, Third edition, Addison-Wesley, Reading, 1993.zbMATHGoogle Scholar
  13. [N]
    D. Neftin, Admissibility and fields relations, Israel Journal of Mathematics 191 (2012), 559–584. DOI: 10.1007/s11856-011-0214-2.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [NP]
    D. Neftin and E. Paran, Admissible groups over two dimensional complete local domains, Algebra & Number Theory 4 (2010), 743–762.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Ne1]
    J. Neukirch, Über das Einbettungsproblem der algebraischen Zahlentheorie, Inventiones Mathematicae 21 (1973), 59–116.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Ne2]
    J. Neukirch, On solvable number fields, Inventiones Mathematicae 53 (1979), 135–164.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [NSW]
    J. Neukirch, A. Schmidt and K. Winberg, Cohomology of Number Fields, Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 323, Springer-Verlag, Berlin, 2008.CrossRefzbMATHGoogle Scholar
  18. [Oe]
    J. Oesterlé, Nombre de Tamagawa et groupes unipotents en caractéristique p, Inventiones Mathematicae 78 (1984), 13–88.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [Sch]
    M. M. Schacher, Subfields of division rings, I, Journal of Algebra 9 (1968), 451–477.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Se]
    J.-P. Serre, Galois Cohomology, Corr. 2 printing; Springer Monographs in Mathematics, Springer, Berlin, 2002.Google Scholar
  21. [So1]
    J. Sonn, Galois groups of global fields of finite characteristic, Journal of Algebra 43 (1976), 606–618.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [So2]
    J. Sonn, Q-admissibility of solvable groups, Journal of Algebra 84 (1983), 411–419.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Ste]
    L. Stern, On the admissibility of finite groups over global fields of finite characteristic, Journal of Algebra 100 (1986), 344–362.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [TT]
    N. Q. Thang and N. D. Tan, On the surjectivity of localization maps for Galois cohomology of unipotent algebraic groups over fields, Communications in Algebra 32 (2004), 3169–3177.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.FB6, MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.Institute of MathematicsHanoiVietnam

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