Prime elements in a localic topos (Q1121977)

The notion of prime filter is studied in the context of sheaves of boolean algebras on a given locale. Unlike the classical case, this notion is no longer equivalent to that of ultrafilter. A rather mild condition, satisfied by all topological spaces, is introduced on the locale and ensures that the classical Stone embedding theorem for boolean algebras holds in the corresponding topos of sheaves. This is achieved by a careful use of the notions of prime filter introduced in the first part of the paper.