Automatic realizations of Galois groups with cyclic quotient of order p n

Ján Mináč[1]; Andrew Schultz[2]; John Swallow[3]

  • [1] Department of Mathematics Middlesex College University of Western Ontario London, Ontario N6A 5B7 CANADA
  • [2] Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 USA
  • [3] Department of Mathematics Davidson College Box 7046 Davidson, North Carolina 28035-7046 USA

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 2, page 419-430
  • ISSN: 1246-7405

Abstract

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We establish automatic realizations of Galois groups among groups M ⋊ G , where G is a cyclic group of order p n for a prime p and M is a quotient of the group ring 𝔽 p [ G ] .

How to cite

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Mináč, Ján, Schultz, Andrew, and Swallow, John. "Automatic realizations of Galois groups with cyclic quotient of order ${p^n}$." Journal de Théorie des Nombres de Bordeaux 20.2 (2008): 419-430. <http://eudml.org/doc/10844>.

@article{Mináč2008,
abstract = {We establish automatic realizations of Galois groups among groups $M\rtimes G$, where $G$ is a cyclic group of order $p^n$ for a prime $p$ and $M$ is a quotient of the group ring $\mathbb\{F\}_p[G]$.},
affiliation = {Department of Mathematics Middlesex College University of Western Ontario London, Ontario N6A 5B7 CANADA; Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801 USA; Department of Mathematics Davidson College Box 7046 Davidson, North Carolina 28035-7046 USA},
author = {Mináč, Ján, Schultz, Andrew, Swallow, John},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {inverse Galois theory; Kummer theory; Galois modules; Galois group; metacyclic group; -group},
language = {eng},
number = {2},
pages = {419-430},
publisher = {Université Bordeaux 1},
title = {Automatic realizations of Galois groups with cyclic quotient of order $\{p^n\}$},
url = {http://eudml.org/doc/10844},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Mináč, Ján
AU - Schultz, Andrew
AU - Swallow, John
TI - Automatic realizations of Galois groups with cyclic quotient of order ${p^n}$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 2
SP - 419
EP - 430
AB - We establish automatic realizations of Galois groups among groups $M\rtimes G$, where $G$ is a cyclic group of order $p^n$ for a prime $p$ and $M$ is a quotient of the group ring $\mathbb{F}_p[G]$.
LA - eng
KW - inverse Galois theory; Kummer theory; Galois modules; Galois group; metacyclic group; -group
UR - http://eudml.org/doc/10844
ER -

References

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  2. H. G. Grundman, T. L. Smith, Automatic realizability of Galois groups of order 16 . Proc. Amer. Math.  Soc. 124 (1996), no. 9, 2631–2640. Zbl0862.12005MR1327017
  3. H. G. Grundman, T. L. Smith, and J. R. Swallow, Groups of order 16 as Galois groups. Exposition. Math. 13 (1995), 289–319. Zbl0838.12004MR1358210
  4. C. U. Jensen, On the representations of a group as a Galois group over an arbitrary field. Théorie des nombres (Quebec, PQ, 1987), 441–458. Berlin: de Gruyter, 1989. Zbl0696.12019MR1024582
  5. C. U. Jensen, Finite groups as Galois groups over arbitrary fields. Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), 435–448. Contemp. Math. 131, Part 2. Providence, RI: American Mathematical Society, 1992. Zbl0780.12004MR1175848
  6. C. U. Jensen, Elementary questions in Galois theory. Advances in algebra and model theory (Essen, 1994; Dresden, 1995), 11–24. Algebra Logic Appl. 9. Amsterdam: Gordon and Breach, 1997. Zbl0939.12001MR1683567
  7. C. U. Jensen, A. Ledet, N. Yui, Generic polynomials: constructive aspects of the inverse Galois problem. Mathematical Sciences Research Institute Publications 45. Cambridge: Cambridge University Press, 2002. Zbl1042.12001MR1969648
  8. T. Y. Lam, Lectures on modules and rings. Graduate Texts in Mathematics 189. New York: Springer-Verlag, 1999. Zbl0911.16001MR1653294
  9. A. Ledet, Brauer type embedding problems. Fields Institute Monographs 21. Providence, RI: American Mathematical Society, 2005. Zbl1064.12003MR2126031
  10. J. Mináč, J. Swallow, Galois module structure of p th-power classes of extensions of degree p . Israel J. Math. 138 (2003), 29–42. Zbl1040.12006MR2031948
  11. J. Mináč, J. Swallow, Galois embedding problems with cyclic quotient of order p . Israel J. Math. 145 (2005), 93–112. Zbl1069.12002MR2154722
  12. J. Mináč, A. Schultz, J. Swallow, Galois module structure of the p th-power classes of cyclic extensions of degree p n . Proc. London Math. Soc. 92 (2006), no. 2, 307–341. Zbl1157.12002MR2205719
  13. D. Saltman, Generic Galois extensions and problems in field theory. Adv. in Math. 43 (1982), 250–283. Zbl0484.12004MR648801
  14. D. Sharpe, P. Vámos, Injective modules. Cambridge Tracts in Mathematics and Mathematical Physics 62. London: Cambridge University Press, 1972. Zbl0245.13001MR360706
  15. W. Waterhouse, The normal closures of certain Kummer extensions. Canad. Math. Bull. 37 (1994), no. 1, 133–139. Zbl0794.12003MR1261568

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