On Kneser extensions
The concepts of \(G\)-Kneser field extension and \(G\)-Cogalois field extension were introduced by \textit{T. Albu} and \textit{F. Nicolae} [J. Number Theory 52, 299-318 (1995; Zbl 0838.12003)], in which they raised the following question: ``Let \(E/F\) be a finite separable field extension such that \(E=F(G)\), with \(F^*\leq G\leq E^*\), \(G/F^*\) a finite group, and such that the canonical maps between the subfields of the extension \(E/F\) and the subgroups of \(G/F^*\) are inverse isomorphisms of lattices. Does there exist another group \(\widetilde G\) with \(F^*\leq\widetilde G\leq E^*\) and \(G/F^*\) a finite group such that \(E/F\) is \(\widetilde G\)-Cogalois?'' A necessary and sufficient condition is given in the paper under review to answer the question positively. Examples in characteristic \(p>0\) and in characteristic 0 are given to show that the answer to the question above is in general negative.
scientific article; zbMATH DE number 1214121
scientific article