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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