Automatic realizations of Galois groups with cyclic quotient of order p n
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, p. 419-430

We establish automatic realizations of Galois groups among groups MG, 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].

Nous établissons des réalisations automatiques de groupes de Galois parmi les groupes MGG est un groupe cyclique d’ordre p n , p premier, et M un groupe quotient de l’anneau 𝔽 p [G].

@article{JTNB_2008__20_2_419_0,
     author = {Min\'a\v c, J\'an and Schultz, Andrew and Swallow, John},
     title = {Automatic realizations of Galois groups with cyclic quotient of order ${p^n}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     pages = {419-430},
     doi = {10.5802/jtnb.635},
     zbl = {1180.12002},
     mrnumber = {2477512},
     zbl = {pre05543170},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_2_419_0}
}
Mináč, Ján; Schultz, Andrew; Swallow, John. Automatic realizations of Galois groups with cyclic quotient of order ${p^n}$. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 419-430. doi : 10.5802/jtnb.635. http://www.numdam.org/item/JTNB_2008__20_2_419_0/

[1] F. Anderson, K. Fuller, Rings and categories of modules. Graduate Texts in Mathematics 13. New York: Springer-Verlag, 1973. | MR 1245487 | Zbl 0301.16001

[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. | MR 1327017 | Zbl 0862.12005

[3] H. G. Grundman, T. L. Smith, and J. R. Swallow, Groups of order 16 as Galois groups. Exposition. Math. 13 (1995), 289–319. | MR 1358210 | Zbl 0838.12004

[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. | MR 1024582 | Zbl 0696.12019

[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. | MR 1175848 | Zbl 0780.12004

[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. | MR 1683567 | Zbl 0939.12001

[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. | MR 1969648 | Zbl 1042.12001

[8] T. Y. Lam, Lectures on modules and rings. Graduate Texts in Mathematics 189. New York: Springer-Verlag, 1999. | MR 1653294 | Zbl 0911.16001

[9] A. Ledet, Brauer type embedding problems. Fields Institute Monographs 21. Providence, RI: American Mathematical Society, 2005. | MR 2126031 | Zbl 1064.12003

[10] J. Mináč, J. Swallow, Galois module structure of pth-power classes of extensions of degree p. Israel J. Math. 138 (2003), 29–42. | MR 2031948 | Zbl 1040.12006

[11] J. Mináč, J. Swallow, Galois embedding problems with cyclic quotient of order p. Israel J. Math. 145 (2005), 93–112. | MR 2154722 | Zbl 1069.12002

[12] J. Mináč, A. Schultz, J. Swallow, Galois module structure of the pth-power classes of cyclic extensions of degree p n . Proc. London Math. Soc. 92 (2006), no. 2, 307–341. | MR 2205719 | Zbl pre05014380

[13] D. Saltman, Generic Galois extensions and problems in field theory. Adv. in Math. 43 (1982), 250–283. | MR 648801 | Zbl 0484.12004

[14] D. Sharpe, P. Vámos, Injective modules. Cambridge Tracts in Mathematics and Mathematical Physics 62. London: Cambridge University Press, 1972. | MR 360706 | Zbl 0245.13001

[15] W. Waterhouse, The normal closures of certain Kummer extensions. Canad. Math. Bull. 37 (1994), no. 1, 133–139. | MR 1261568 | Zbl 0794.12003