On Cebotarev Sets

Abstract

where L|K is a finite Galois extension with Galois group G(L|K), σ ∈ G(L|K) and ( L|K P ) denotes the Frobenius automorphism with respect to P, P an arbitrary extension of p to L. The precise definition of the topology TK of PK is slightly more complicated (see §2) since we want that the natural map φK′|K : (PK′ , TK′)−→(PK , TK), P → P ∩K, is continuous if K ′|K is a finite extension. We will show that (PK , TK) is a strongly zero-dimensional (and so totally disconnected) Hausdorff space with countable base, and so metrizable, hence normal and completely regular (and not discrete). In particular, every point of (PK , TK) has a base of neighbourhoods consisting of both open and closed sets. Furthermore we will prove the following theorem (2.8)