Kneser field extensions with cogalois correspondence
Let \(K\) be a subfield of a field \(L\), and let \(K^*\) (resp. \(L^*)\) be the multiplicative group of nonzero elements of \(K\) (resp. \(L)\). The field \(L\) is a radical extension of \(K\) if there is a subgroup \(G\) of \(L^*\) such that \(G\) contains \(K^*\), \(G/K^*\) is a torsion group, and \(L = K(G)\). If, in addition, \([G : K^*]\) and \([L : K]\) are finite and equal, the authors say that \(L\) is a \(G\)-Kneser extension of \(K\). These conditions were used by \textit{K. Hoechsmann} [Proc. Am. Math. Soc. 14, 768-776 (1963; Zbl 0224.13001)] to characterize those purely inseparable field extensions which are now called modular. Equivalent conditions are that \(L\) is a comodule algebra, with respect to the diagonal map of the group algebra of \(G/K^*\) over \(K\), which is a Hopf-Galois extension of \(K\). When \(G/K^*\) is the torsion group of \(L^*/K^*\), a \(G\)-Kneser extension is called cogalois by \textit{C. Greither} and \textit{D. K. Harrison} [J. Pure Appl. Algebra 43, 257-270 (1986; Zbl 0607.12015)]. There is a ``Cogalois connection'' between subgroups of \(G/K^*\) and certain intermediate fields between \(K\) and \(L\) for a \(G\)-Kneser extension. Now assume that \(L\) is a separable extension of \(K\). After using a result of \textit{M. Kneser} [Acta Arith. 26, 307-308 (1975; Zbl 0314.12001)] to determine when a radical extension is \(G\)-Kneser and when it is cogalois, the authors consider when the cogalois connection involves all intermediate fields between \(K\) and \(L\). They prove the necessity and sufficiency of the condition that any \(p\)-th root of unity, for \(p\) an odd prime dividing the exponent of \(G/K^*\) or \(p = 4\) if 4 divides the exponent of \(G/K^*\), which is present in \(L\) belongs to \(K\). They conclude the paper with a study of field extensions which are both cogalois and Galois, included among which are Kummer extensions.
scientific article; zbMATH DE number 770125