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Galois embedding problems with cyclic quotient of orderp

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Abstract

LetK/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable\(\mathbb{F}_p \left[ {Gal\left( {K/F} \right)} \right] - modules\). For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.

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References

  1. A. Albert,On cyclic fields, Transactions of the American Mathematical Society37 (1935), 454–462.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. G. Grundman, T. L. Smith and J. R. Swallow,Groups of order 16 as Galois groups, Expositiones Mathematicae13 (1995), 289–319.

    MATH  MathSciNet  Google Scholar 

  3. C. Jensen, A. Ledet and N. Yui,Generic polynomials: constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications 45, Cambridge University Press, Cambridge, 2002.

    MATH  Google Scholar 

  4. T.-Y. Lam,Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1999.

    MATH  Google Scholar 

  5. R. Massy,Construction de p-extensions Galoisiennes d’un corps de caractéristique différente de p, Journal of Algebra109 (1987), 508–535.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Mináč and J. Swallow,Galois module structure of pth-power classes of extensions of degree p, Israel Journal of Mathematics138 (2003), 29–42.

    Article  MathSciNet  Google Scholar 

  7. D. Saltman,Generic Galois extensions and problems in field theory, Advances in Mathematics43 (1982), 250–283.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. Waterhouse,The normal closures of certain Kummer extensions, Canadian Mathematical Bulletin37 (1994), 133–139.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Middlesex College, University of Western Ontario, N6A 5B7, London, Ontario, Canada

    Ján Mináč

  2. Department of Mathematics, Davidson College, Box 7046, 28035-7046, Davidson, NC, USA

    John Swallow

Authors
  1. Ján Mináč
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  2. John Swallow
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Corresponding author

Correspondence to Ján Mináč.

Additional information

Research supported in part by the Natural Sciences and Engineering Research Council of Canada grant R0370A01, as well as by the special Dean of Science Fund at the University of Western Ontario.

Supported by the Mathematical Sciences Research Institute, Berkeley.

Research supported in part by National Security Agency grant MDA904-02-1-0061.

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Mináč, J., Swallow, J. Galois embedding problems with cyclic quotient of orderp . Isr. J. Math. 145, 93–112 (2005). https://doi.org/10.1007/BF02786686

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  • DOI: https://doi.org/10.1007/BF02786686

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