Descent and Galois parallelograms. (Descente et parallélogramme galoisiens.) (French) Zbl 0944.12002

Given a Galois extension \(E/K\) and an algebraic extension \(K/J\), the problem of Galois descent is to find conditions under which there exists a subfield \( D\subseteq E\) which is Galois over \(J\) such that \(D \cap K = J \) and \(D K = E\).
For an odd prime number \(p\), let \(D/J\) be a Galois \(p\)-extension with Galois group \(G\) such that \(J \cap \mu _p = \{1\}\), where \(\mu _p\) denotes the group of \(p\)-th roots of unity. Using the Galois descent notion and induced Galois parallelograms the authors construct all extensions \(D/J\) such that the Frattini subgroup of \(G\) is of order \(p\). In particular, they give an explicit description of a primitive element of all cyclic non-Kummerian extensions of degree \(p\) and \(p ^2\).


12F10 Separable extensions, Galois theory
11R32 Galois theory
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[1] Brattström, G., On p-groups as Galois groups. Math. Scand.65 (1989), 165-174. · Zbl 0707.12001
[2] Bruen, A.A., Jensen, C.U. and Yui, N., Polynomials with Frobenius groups of prime degree as Galois groups II. J. Number Theory24 (1986), 305-359. · Zbl 0598.12009
[3] Hoechsmann, K., Zum Einbettungsproblem. J. reine angew. Math.229 (1968), 81-106. · Zbl 0185.11202
[4] Huppert, B., Endliche Gruppen I, 2nd ed. Springer-Verlag, Berlin, 1983. · Zbl 0217.07201
[5] Karpilovsky, G., Topics in Field Theory. North-Holland Mathematics Studies155, Amsterdam, 1989. · Zbl 0662.12023
[6] Massy, R., Construction de p-extensions galoisiennes d’un corps de caractéristique différente de p. J. Algebra109 (1987), 508-535. · Zbl 0625.12011
[7] Massy, R. et Monier-Derviaux, S., Parallélogrammes galoisiens. J. Algebra, à paraître. · Zbl 0973.12002
[8] Monier, S., Descente de p-extensions galoisiennes kummériennes. Math. Scand.79 (1996), 5-24. · Zbl 0876.12004
[9] Monier-Derviaux, S., Le Problème de la Descente Galoisienne Finie. Thèse de Doctorat, Univ. Valenciennes, 1997.
[10] Wôjcik, J., Criterion for a field to be abelian. Colloq. Math.68 (1995), 187-191. · Zbl 0827.11063
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