Massy, Richard; Monier-Derviaux, Sylvie Descent and Galois parallelograms. (Descente et parallélogramme galoisiens.) (French) Zbl 0944.12002 J. Théor. Nombres Bordx. 11, No. 1, 161-172 (1999). 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 \(p ^2\). Reviewer: Gabriel D.Villa-Salvador (México) Cited in 2 ReviewsCited in 3 Documents MSC: 12F10 Separable extensions, Galois theory 11R32 Galois theory Keywords:Galois descent; Galois parallelograms; cyclic non-Kummerian extensions × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML EMIS References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.