Abstract:

In Proposition I of "Mémoire sur les conditions de résolubilité des équations par radicaux" Galois considers the splitting field A of a polynomial with rational coefficients. Given any intermediate extension X, he proves that the action of the galois group Gal(A/ℚ) on the set of morphisms [X, A] is transitive, and that X is the fixed field of its galois group Gal(A/X). In this article we first state and prove a (dual) categorical formulation of these statements, which turns out to be a theorem about atomic sites with a representable point. These abstract developments correspond exactly to Classical Galois Theory. We then consider the same situation, but with the point no longer representable. It determines a proobject and it becomes (tautologically) prorepresentable. Mutatis mutandis, we state and prove the prorepresentable version of the categorical formulation of Galois theorems. In this case the classical group of automorphisms has to be replaced by the localic group of automorphisms. These developments form the content of a theory that we call Localic Galois Theory. From our results it immediately follows the theorem: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can he taken to be the locale of automorphisms of the point. The hard implication in this equivalence was first proved in print in Joyal and Tierney (Mem. Amer. Math. Soc. 151 (1984)), Theorem 1, Section 3, Chapter VIII, and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales. Our proof of this theorem is completely independent of descent theory and of any other result in Joyal and Tierney (1984). The theorem follows as a straightforward consequence of our direct generalization of the fundamental results of Galois. We explain and give the necessary definitions to understand this paper without previous knowledge of the theory of locales. Other than this, our principal results (Theorems 2.2 and 2.3) require only elementary notions of category theory. © 2003 Elsevier Science (USA). All rights reserved.