The monograph presents the current state of Galois theory and consists of seven chapters and an appendix. In Chapter 1 an elementary presentation of the classical Galois theory is given.
Chapter 2, inspired by the work of Grothendieck, proves that the classical Galois theorem of Chapter 1 is a local segment of a more general equivalence of categories. Given a finite dimensional Galois field extension \(K\subseteq L\), the Galois theorem asserts now that the category of finite dimensional \(K\)-algebras split by \(L\) is equivalent to the category of finite \(\text{Gal}[L:K]\)-sets, that is, finite sets provided with an action of the Galois group. The classical Galois theorem of Chapter 1 is recaptured by observing that the subgroups \(G\subseteq \text{Gal}[L:K]\) correspond bijectively with the quotients of the Galois group in the category of \(\text{Gal}[L:K]\)-sets; via the equivalence of categories, these are in bijection with the split algebras \(K\subseteq A\subseteq L\), which turn out to be the intermediate fields.
In Chapter 3, the case of an arbitrary Galois extension of fields \(K\subseteq L\), not necessarily finite dimensional, is handled. In that case, the Galois group \(\text{Gal}[L:K]\) comes naturally equipped with a profinite topology, i.e., the structure of a compact Hausdorff space whose topology admits a base of closed-open subsets. The classical version of the Galois theorem asserts now that the closed subgroups of the profinite Galois group classify the intermediate extensions \(K\subseteq M\subseteq L\). The Galois theorem of Grothendieck extends in an analogous way, yielding now an equivalence between the category of \(K\)-algebras split by \(L\) and the category of profinite \(\text{Gal}[L:K]\)-spaces, with continuous action of the Galois group.
Chapter 4 is devoted to developing the Galois theory of rings. New ingredients, namely, the Pierce spectrum of the ring and the Galois descent morphism \(\sigma: R\to S\) are introduced. The Galois group is replaced by a Galois groupoid. The Galois theorem for rings asserts that the category of internal presheaves on the Galois groupoid is equivalent to the category of \(R\)-algebras split by \(\sigma\).
Chapter 5 is the core of the book. The categorical context in which a general Galois theorem holds is formalized and some applications are given. The Pierce spectrum functor between the category of rings and that of profinite spaces is replaced by an arbitrary functor between arbitrary categories. The assumptions required to infer a Galois theorem proving an equivalence of categories between “split algebras” for this adjunction and the internal presheaves on some internal Galois groupoid are exhibited.
Chapter 6 focuses on the notion of covering map of topological spaces. If \(B\) is connected and locally connected, and has a universal covering \((E,p)\), then all connected coverings of \(B\) are quotients of \((E,p)\) and there is a bijection between them and the subgroups of the automorphism group \(\operatorname{Aut} (E,p)\). That bijection is constructed precisely a the standard Galois correspondence for separable field extensions, but in the dual category \((\text{Top}/B)^{op}\) of bundles over \(B\).
Finally, in Chapter 7, the authors show that it is possible to get a Galois theorem in the general context of descent theory without necessarily a Galois assumption. This yields in particular a Galois theorem for every field extension \(K\subseteq L\) without any further assumption. The Galois group or the Galois groupoid must now be replaced by the more general notion of precategory. But in some cases of interest, even without any Galois assumption, the Galois precategory turns out to be again a groupoid. For example, the Galois theorem for toposes, due to Joyal and Tierney, enters the context of the generalized theory without any Galois assumption. It asserts that every topos is equivalent to a category of étale presheaves on an open étale groupoid.