Galois theory and theaters of action in a topos

Abstract

Constructing a Galois theory for geometric fields in a topos presents certain difficulties. If K E F is a Galoisian field extension in a topos 5Z, then one can construct the internal group of K-automorphisms of F but it might be trivial even when K f F (see Example 6.6). In the topos of presheaves over a category C, a Galois group appears to be a presheaf over Cop. There are similar difficulties with splitting a polynomial over a field K in a topos. If we are splitting x2 + 1 over K in the topos of Sets, then we usually use a dichotomous procedure: if x2+ 1 is irreducible we construct K[i], if .r* + 1 is reducible we use K. This procedure is problematic for topoi since irreducibility is not a coherent condition (i.e., not geometric in the sense of [3]). A procedure for splitting polynomials is given in Section 3. See Example 6.3 also. Of course the field containing the added roots lives in a new topos, and this is an instance of the Cole Spectrum (see [l, 3,7]).