Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T18:03:40.246Z Has data issue: false hasContentIssue false
  • English
  • Français

The Normal Closures of Certain Kummer Extensions

Published online by Cambridge University Press:  20 November 2018

William C. Waterhouse
William C. Waterhouse*
Affiliation:
Department of Mathematics The Pennsylvania State University University Park, Pennsylvania 16802 U.S.A.
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 F be a field containing a primitive p-th root of unity, let K / F be a cyclic extension with group 〈σ〉 of order pn, and choose a in K. This paper shows how the Galois group of the normal closure of K(a1/p) over F can be determined by computations within K. The key is to define a sequence by applying the operation x ↦ σ(x)/x repeatedly to a. The first appearance of a p-th power determines the degree of the extension and restricts the Galois group to one or two possibilities. A certain expression involving that p-th root and the terms in the sequence up to that point is a p-th root of unity, and the group is finally determined by testing whether that root is 1. When (σ(a)/a G Kp, the results reduce to a theorem of A. A. Albert on cyclic extensions.

Keywords

12F10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 1 , 01 March 1994 , pp. 133 - 139
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Albert, A. A., On normal Kummer fields over a non-modular fieldTrans. Amer. Math. Soc. 36(1934), 885–892.Google Scholar
2. Albert, A. A., On cyclic fields. Trans. Amer. Math. Soc. 37( 1935 ), 452–462.Google Scholar
3. Albert, A. A., Modern Higher Algebra, University of Chicago Press. Chicago, 1937.Google Scholar
4. Artin, E., Galois Theory, Notre Dame Math. Lectures (2), Notre Dame, 1959.Google Scholar
5. Bourbaki, N., Algèbre, Chapitres 4-7, Masson, Paris, 1981.Google Scholar
6. Brauer, R., Uber die Konstruklion der Schiefkorper, die von endlichem Rang in bezug auf ein gegebenes Zentrum sind, J. Reine Angew. Math. 168( 1932), 44-64; reprinted in Collected Papers 1, 121–141.Google Scholar
7. Damey, P. and Pay, J.-J. an. Existence et construction des extensions galoisiennes et non-abèliennes de degré8 d'un corps de characteristic différente de 2, J. Reine Angew. Math. 244(1970), 37–54.Google Scholar
8. Hall, M., Jr., The Theory of Groups, Macmillan, New York, 1959.Google Scholar
9. Hasse, H., Existenz unci Mannigfaltigkeit abelscher Algebren mit vorgegebener Galoisgruppe über einem Teilkörperdes Grundkorper, Math. Nachr. 1(1948), 40–61.Google Scholar
10. Ishanov, V. V., On the semidirect imbedding problem with nilpoteht kernel, Izv. Akad. Nauk SSSR Ser. Mat. 40(1976), 3–25; Math USSR-Izv. 10(1976), 1–23.Google Scholar
11. Jacobson, N., Basic Algebra I, Freeman, San Francisco, 1974.Google Scholar
12. Jacobson, N., Basic Algebra II, Freeman, San Francisco, 1980.Google Scholar
13. Kiming, I., Explicit classifications of some 2-extensions of a field of characteristic different from2, Canad. J. Math. 42(1990), 825–855.Google Scholar
14. Payan, J.-J., Critère de décomposition d'une extension de Kummer sur un sous-corps du corps de base, Ann. Sci. École Norm. Sup. (4) 1(1968), 445–458.Google Scholar
15. Serre, J.-R, Corps locaux, Hermann, Paris, 1962.Google Scholar
16. Shafarevich, I. R., Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR Ser. Mat. 18(1954), 525–578; Amer. Math. Soc. Translations (II) 4(1956), 185–237.Google Scholar
17. Vaughan, T. P., The normal closure of a quadratic extension of a cyclic quartic field, Canad. J. Math 43(1991),1086–1097.Google Scholar
18. Vaughan, T. P., Constructing quaternionicfields, Glasgow Math. J. 34(1992), 43–54.Google Scholar
19. Ware, R., A note on the quaternion group as Galois group, Proc. Amer. Math. Soc. 108(1990), 621–625.Google Scholar