Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T15:45:50.399Z Has data issue: false hasContentIssue false
  • English
  • Français

On 2-Groups as Galois Groups

Published online by Cambridge University Press:  20 November 2018

Arne Ledet
Arne Ledet*
Affiliation:
Matematisk Institut Universitetsparken 5 DK-2100 Copenhagen Ø Denmark e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

Keywords

12F12
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1253 - 1273
Copyright
Copyright © Canadian Mathematical Society 1995

References

[As] Aschbacher, M., Finite Group Theory, Cambridge Stud. Adv. Math. 10, Cambridge Univ. Press, 1986.Google Scholar
[Br] Brauer, R., Über die Konstruktion der Schiefkörper, die von endlichem Rang in bezug aufein gegebenes Zentrum sind, J. Reine Angew. Math. 168(1932), 44–64.Google Scholar
[Fr] Fröhlich, A., Orthogonal representations of Galois groups, StiefeI- Whitney classes and Hasse-Witt invariants,, J. Reine Angew. Math. 360(1985), 84–123.Google Scholar
[G&S]Grundman, H.G. and Smith, T.L., Automatic readability of Galois groups of order 16, (1994), preprint.Google Scholar
[GS&S] Grundman, H.G., Smith, T.L. and Swallow, J.R., Groups of order 16 as Galois groups, (1994), preprint.Google Scholar
[Ho] Hoechsmann, K., Zum Einbettungsproblem, J. Reine Angew. Math. 229(1968), 81–106.Google Scholar
[Ja] Jacobson, N., Basic Algebra II, W.H. Freeman and Company, New York, 1989.Google Scholar
[J1] Jensen, C.U., On the representations of a group as a Galois group over an arbitrary field. In: Théorie des nombres Number Theory, (eds. De, J.-M. Koninck and Levesque, C.), Walter de Gruyter, 1989.441-458.Google Scholar
[J2] Jensen, C.U., Finite groups as Galois groups over arbitrary fields. In: Contemp. Math. 131, Proceedings of the international conference of algebra 1989, part 2, Amer. Math. Soc, 1992. 435—448.Google Scholar
[J&Y] Jensen, C.U. and Yui, N., Quaternion Extensions. In: Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, Tokyo, 1987. 155-182.Google Scholar
[Ki] Kiming, I., Explicit Classifications of some 2-Extensions of a Field of Characteristic different from 2,, Canad. J. Math. 42(1990), 825–855.Google Scholar
[K&L] Kuyk, W. and Lenstra, H.W., Jr., Abelian extensions of arbitrary fields, Math. Ann. 216(1975), 99–104.Google Scholar
[La] Lam, T.Y., The Algebraic Theory of Quadratic Forms, W.A.Benjamin, Reading, Massachusetts, 1973.Google Scholar
[Lo] Lorenz, F., Einführung in die Algebra II, B.I. Wissenschaftsverlag, Mannheim, 1990.Google Scholar
[Me] Merkurjev, A., On the norm residue symbol of degree, 2, Soviet Math. Dokl. 24(1981), 546–551.Google Scholar
[M&S] Mináč, J. and Smith, T.L., A characterization ofC-fields via Galois groups, J. Algebra 137( 1991 ), 1–11.Google Scholar
[Sc] Schneps, Leila, Explicit Realisations of Subgroups of GLüF?) as Galois Groups, J. Number Theory 39(1991), 5–13.Google Scholar
[Se] Serre, J.-P., A Course in Arithmetic, Graduate Texts in Math. 7, Springer-Verlag, 1973.Google Scholar
[Sm] Smith, T., Extra-special groups of order 32 as Galois groups, Canad. J. Math. 46(1994), 886–896.Google Scholar
[Wa] Ware, R., Automorphisms of Pythagorean Fields and their Witt Rings, Comm. Algebra 17(1989), 945–969.Google Scholar
[Wh] Whaples, G., Algebraic extensions of arbitrary fields, Duke Math. J. 24(1957), 201–204.Google Scholar
[Wi] Witt, E., Konstruktion von galoisschen Körpern der Characteristik p zu vorgegebener Gruppe der Ordnung pf,, J. Reine Angew. Math. 174(1936), 237–245.Google Scholar