## 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 $$p ^2$$.

### MSC:

 12F10 Separable extensions, Galois theory 11R32 Galois theory
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### References:

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