Kneser Field Extensions with Cogalois Correspondence

Abstract

A held extension K ⊆ L is said to be an extension with G-Cogalois correspondence if there exists a canonical lattice isomorphism between the lattice of all subextensions of K ⊆ L and the lattice of all subgroups of a certain group G. Such extensions are, cf. Greither and Harrison ( J. Pure Appl. Algebra 43 (1986), 257- 270), the Cogalois extensions and the neat presentations, as well as, by a well-known result, the classical finite Kummer extensions. However, the proofs in Greither and Harrison of these facts, especially for neat presentations, are not natural enough and rather complicated. The aim of this paper is to provide a very simple and unitary approach of all these cases, by placing them in a more general setting. Thus, the class of G-Cogalois extensions is introduced and completely characterized in terms of a certain kind of "local" purity, called n-purity, inside the class of so called G-Kneser extensions. A nice criterion concerning the evaluation of the degrees of finite radical extensions, due to Kneser ( Acta Arith. 26 (1975), 307-308), which originated our definition of G-Kneser extension is the main tool in all our investigation. A series of results due to Albu, Barrera-Mora et al., Gay and Vélez, Greither and Harrison are simplified, unified and extended.