Dear All, The following recent message of Tobias Schroeder

Hello, all introductions to field and Galois theory I've found are written in a "classical" way, i.e. making not much use of categorical notions. A lot of computation is done where someone who is "categorical minded" has the feeling that the results could be established in a more comprehensible and clear way by category theory. -- Does somebody have a reference to a short and good introduction to field and Galois theory from a categorical viewpoint?

Thanks

Tobias Schroeder

was already answered by Francis Borceux, who has beautifully written most of the book "Galois theories". I would like to add: I would say, "Field and Galois theory" sounds too general. For instance any such introduction should include a lot of Group theory and polynomials, which of course would look much nicer if various parts of Category theory were involved - but this is a very long story! So, let me replace "Field and Galois theory" by just "The fundamental theorem of Galois theory" - and call it GFT for short. The standard formulation of GFT includes the following assertions about a finite Galois extension E/K with the Galois group G = Gal(E/K): GFT1: The opposite lattice of subextensions of E/K is isomorphic to the lattice of subgroups of G; under this isomorphism a subextension F/K corresponds to the subgroup Gal(E/F) = {g in G: ga = a for all a in F}, and therefore a subgroup H in G corresponds to {a in E: ga = a for all g in H}. GFT2: A subextension F/K of E/K is normal (equivalently, Galois) if and only if its corresponding subgroup Gal(E/F) is normal, and if this is the case, then Gal(F/K) is canonically isomorphic to the quotient group. Moreover, every K-homomorphism of subextensions of E/K extends to a K-automorphism of E. Unfortunately even today all books in Algebra give only this kind of formulation. I think actually the right name for it is not "standard" but "prehistoric" - since more than 40 years ago Chevalley and Grothendieck understood that it is a straightforward consequence of the following simple and nice formulation: Grothendieck's GFT restricted: The category of subextensions of E/K (with morphisms all K-homomorphisms) is equivalent to the category of transitive G-sets, where E/K and G are as above. Moreover, one does not really want what I called "restricted", and then the right formulation becomes: Grothendieck's GFT: The category of K-algebras split over E/K is equivalent to the category of finite G-sets. Here a K-algebra A is said to be split over E/K if its tensor product over K with E is isomorphic to the Cartesian product of a finite number of copies of E; note that A is split over E/K if and only if it is itself isomorphic to the Cartesian product of a finite number of subextensions of E/K. There are many theorems similar or more general then this, proved by Chevalley and Grothendieck themselves, by A. R. Magid, M. Barr and R. Diaconescu, and others. In 1984 I realized that there is a purely categorical formulation and a purely categorical proof - before that the topos-theoretic level was considered as the most general, although there was no topos-theoretic extension of Magid's theorem. And what I call now Categorical Galois theory - let us say CGT for short - has important examples very far from Grothendieck and topos theory. One of them, studied in joint work with G. M. Kelly is of what we called generalized central extensions in universal algebra. CGT actually uses very simple category theory (pullbacks, adjoint functors, monadicity, internal category actions), but after many attempts I found it very difficult to explain it to "non-category-theorists" - I would say, simply because most of them do not believe that General Category theory can have non-trivial applications! A further generalization of CGT to so-called variable categories was developed in joint work with D. Schumacher and R. H. Street. In some sense it includes Street's theory of torsors, Joyal - Tierney's theorem on geometric morphisms of toposes, and Tannaka duality. George Janelidze