Separable algebraic closure in a topos

Abstract

In this paper we construct the separable algebraic closure of any field in a topos 8. (A separable closure is a field extension in which every separable polynomial splits and which is generated by the roots of these polynomials. P is a separable polynomial if P and its derivative P’ are relatively prime.) The separable closure lives in a new topos, over b, and has the universal property which makes it a spectrum in the sense of Cole, see [l], [9] and [4, Theorem 6.581. Recall that if 8 is the topos of Sets there are two ways to construct this spectrum. One is to use the &ale topos which is the home of the generic separable closure of an arbitrary commutative ring in Sets (see [3,12]). As shown by Hakim [3, pp. 77-841, this construction can be applied within any Grothendieck topos by setting up the &ale topology (object by object on the defining site of the topos). Alternatively, we can describe the generic separable closure of a field as a field extension in the topos of continuous G-Sets where G is the profinite Galois group. In this paper, we generalize the profinite Galois group approach. We feel that the value of this paper lies not so much in the alternative, more internal, construction of the separable closure but in the topos theoretic structure we develop along the way. We define a profinite group and more generally profinite groupoid, and profinite category in any topos. If I-is a profinite category in 8 then we construct the topos gr of continuous r-actions. We generalize the profinite Galois group of the algebraic closure of a field. In general it is a connected profinite groupoid. As shown in Section 3, below, it is Morita equivalent to a profinite group precisely when the separable closure of K in 8 can be constructed within the topos 8 itself. For example, any field in Sets has an algebraic closure in Sets so the profinite Galois groupoid is effectively equivalent to a profinite group. When there does exist a separable closure of K in 8 itself then the Galois groupoid is equivalent to the profinite group constructed in [6].