On the Krull Galois theory for non-algebraic extension fields

Abstract

The Krull Galois theory for infinite separable normal extensions is generalized in this note to non-algebraic extensions. For any extension fieldEof a fieldKit is shown that the Galois groupGcan be given a translation invariant topology such that the closed subgroups are precisely the subgroups that figure in a Galois correspondence. For extension fieldsE/Ksuch thatE/Kis of finite transcendence degree and such thatEis Galois over each intermediate field the topology turns out to be compact and we have a Galois correspondence in the Krull fashion. For infinite transcendence degree extensions the Galois correspondence remains but compactness is lost. The topology coincides with the Krull topology in the case of algebraic extensions. Further properties of the topology are also studied.