Galois parallelograms
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.
scientific article; zbMATH DE number 1340611
scientific article