LATTICE-ISOMORPHIC GROUPS, AND INFINITE ABELIAN <i>G</i>-COGALOIS FIELD EXTENSIONS

Abstract

The aim of this paper is to provide a proof of the following result claimed by Albu (Infinite field extensions with Galois-Cogalois correspondence (II), Revue Roumaine Math. Pures Appl. 47 (2002), to appear): The Kneser group Kne(E/F) of an Abelian G-Cogalois extension E/F and the group of continuous characters Ch(Gal(E/F)) of its Galois group Gal(E/F) are isomorphic (in a noncanonical way). The proof we give in this paper explains why such an isomorphism is expected, being based on a classical result of Baer (Amer. J. Math. 61 (1939), 1-44) devoted to the existence of group isomorphisms arising from lattice isomorphisms of their lattices of subgroups.