Galois parallelograms. (Parallélogrammes galoisiens.) (French) Zbl 0973.12002

Massy, Richard; Monier-Derviaux, Sylvie

J. Algebra 217, No. 1, 229-248 (1999).

Let \(K/J\) be an algebraic extension of fields and \(N/K\) be a Galois extension. A natural question is, does there exist a field \(L\), \(J\subseteq L \subseteq N\) such that \(L/J\) is a Galois extension with \(L\cap K = J\) and \(N = KL\)? In the case that such \(L\) exists, it is said that \(N/K\) descends to \(J\) and that \(L/J\) is a descent of \(N/K\). We have \(\text{Gal}(N/K) \cong \text{Gal}(L/J)\).
The authors generalize the notion of Galois extension by defining Galois parallelograms as above and considering \(K/J\) also a Galois extension. A Galois extension is just a “flat” parallelogram, that is, when \(K=J\) and \(N=L\). We may consider the theory of Galois parallelograms as two dimensional Galois theory.
After stating the principal general and arithmetic properties of Galois parallelograms, generalizing some of their previous results [J. Théor. Nombres Bordx. 11, No. 1, 161-172 (1999; Zbl 0944.12002)], the authors define the Galois group of a Galois parallelogram and prove the usual properties of classical Galois theory.
The last section is dedicated to prove the general properties of Galois parallelograms stated at the beginning of the paper.

Reviewer: Gabriel D.Villa-Salvador (México)

MSC:

12F10	Separable extensions, Galois theory
11R32	Galois theory

Keywords:

Galois descent; Galois parallelogram; two-dimensional Galois theory

Citations:

Zbl 0944.12002

References:

[1]	Brattström, G., On \(p\)-groups as Galois groups, Math. Scand., 65, 165-174 (1989) · Zbl 0707.12001
[2]	Lang, S., Algebra (1995), Addison-Wesley: Addison-Wesley Reading
[3]	Ledet, A., Subgroups of Hol \(Q_8\) as Galois groups, J. Algebra, 181, 478-506 (1996) · Zbl 0849.12007
[4]	Massy, R., Construction de \(p\)-extensions galoisiennes d’un corps de caractéristique différente de \(p\), J. Algebra, 109, 508-535 (1987) · Zbl 0625.12011
[7]	Monier, S., Descente de \(p\)-extensions galoisiennes kummériennes, Math. Scand., 79, 5-24 (1996) · Zbl 0876.12004
[8]	Serre, J.-P., Corps Locaux (1980), Hermann: Hermann Paris
[9]	Swallow, J. R., Central \(p\)-extensions of \((p,p\),…,\(p)\)-type Galois groups, J. Algebra, 186, 277-298 (1996) · Zbl 0870.12005