Field Theoretic Cogalois Theory via Abstract Cogalois Theory

Abstract Cogalois Theory for arbitrary profinite groups was initiated by T. Albu and Ş.A. Basarab [An Abstract Cogalois Theory for profinite groups, J. Pure. Appl. Algebra 200 (2005) 227–250]. The aim of this paper is twofold: firstly, to present the abstract group theoretic versions of various types of Kummer field extensions, and secondly, to show how some basic results of the (Field Theoretic) Cogalois Theory can be very easily deduced from their abstract versions.