𝑝-groups have unbounded realization multiplicity

Berg, Jen; Schultz, Andrew

doi:10.1090/S0002-9939-2014-11967-8

$p$-groups have unbounded realization multiplicity
HTML articles powered by AMS MathViewer

by Jen Berg and Andrew Schultz PDF

Proc. Amer. Math. Soc. 142 (2014), 2281-2290 Request permission

Abstract:

In this paper we interpret the solutions to a particular Galois embedding problem over an extension $K/F$ satisfying $\operatorname {Gal}(K/F) \simeq \mathbb {Z}/p^n\mathbb {Z}$ in terms of certain Galois submodules within the parameterizing space of elementary $p$-abelian extensions of $K$; here $p$ is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of $p$-groups whose realization multiplicity is unbounded.

References

Gudrun Brattström, On $p$-groups as Galois groups, Math. Scand. 65 (1989), no. 2, 165–174. MR 1050862, DOI 10.7146/math.scand.a-12276
Helen G. Grundman, Tara L. Smith, and John R. Swallow, Groups of order $16$ as Galois groups, Exposition. Math. 13 (1995), no. 4, 289–319. MR 1358210
C. U. Jensen, On the representations of a group as a Galois group over an arbitrary field, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 441–458. MR 1024582
C. U. Jensen, Finite groups as Galois groups over arbitrary fields, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 435–448. MR 1175848, DOI 10.24033/msmf.303
C. U. Jensen, Elementary questions in Galois theory, Advances in algebra and model theory (Essen, 1994; Dresden, 1995) Algebra Logic Appl., vol. 9, Gordon and Breach, Amsterdam, 1997, pp. 11–24. MR 1683567
Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR 1969648
C. U. Jensen and A. Prestel, Unique realizability of finite abelian $2$-groups as Galois groups, J. Number Theory 40 (1992), no. 1, 12–31. MR 1145851, DOI 10.1016/0022-314X(92)90025-K
C. U. Jensen and A. Prestel, How often can a finite group be realized as a Galois group over a field?, Manuscripta Math. 99 (1999), no. 2, 223–247. MR 1697215, DOI 10.1007/s002290050171
W. Kuyk and H. W. Lenstra Jr., Abelian extensions of arbitrary fields, Math. Ann. 216 (1975), no. 2, 99–104. MR 424772, DOI 10.1007/BF01432536
Ivo M. Michailov, Groups of order 32 as Galois groups, Serdica Math. J. 33 (2007), no. 1, 1–34. MR 2313793
Ivo M. Michailov, Four non-abelian groups of order $p^4$ as Galois groups, J. Algebra 307 (2007), no. 1, 287–299. MR 2278055, DOI 10.1016/j.jalgebra.2006.05.021
Ján Mináč and John Swallow, Galois module structure of $p$th-power classes of extensions of degree $p$, Israel J. Math. 138 (2003), 29–42. MR 2031948, DOI 10.1007/BF02783417
Ján Mináč and John Swallow, Galois embedding problems with cyclic quotient of order $p$, Israel J. Math. 145 (2005), 93–112. MR 2154722, DOI 10.1007/BF02786686
Ján Mináč, Andrew Schultz, and John Swallow, Galois module structure of $p$th-power classes of cyclic extensions of degree $p^n$, Proc. London Math. Soc. (3) 92 (2006), no. 2, 307–341. MR 2205719, DOI 10.1112/S0024611505015479
Ján Mináč, Andrew Schultz, and John Swallow, Automatic realizations of Galois groups with cyclic quotient of order $p^n$, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 419–430 (English, with English and French summaries). MR 2477512
A. Schultz, Parameterizing solutions to any Galois embedding problem over $\mathbb {Z}/p^n\mathbb {Z}$ with elementary $p$-abelian kernel. Preprint.
William C. Waterhouse, The normal closures of certain Kummer extensions, Canad. Math. Bull. 37 (1994), no. 1, 133–139. MR 1261568, DOI 10.4153/CMB-1994-019-4
G. H. Wenzel, Note on G. Whaples’ paper “Algebraic extensions of arbitrary fields”, Duke Math. J. 35 (1968), 47. MR 220709
G. Whaples, Algebraic extensions of arbitrary fields, Duke Math. J. 24 (1957), 201–204. MR 85228
E. Witt, Konstruktion von galoisschen Körpern der Charakteristik $p$ zu vorgegebener Gruppe der Ordnung $p^f$, J. Reine Angew. Math. 174 (1936), 237–245.