This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin should still be in full occupation of this particular ecological niche. To give context, I find that the table of contents of Algèbre et théories galoisiennes by Douady and Douady (at www.gbv.de/dms/goettingen/349204683.pdf) gives more of an idea of Galois theory as I might understand its scope (excluding perhaps the final chapter on dessins d'enfant, see http://en.wikipedia.org/wiki/Dessin_d%27enfant, but even that is debatable).

First point would be this: the Douady book is clearly heavily influenced by the work of Grothendieck, particularly SGA1. So one question is whether SGA1 is really the new orthodoxy on Galois theory, and deserves to be recognised as such? To recap on what happened (c.1960),

*Galois groups are pro-finite groups G, with finite Galois groups being a special case;

*Galois theory is reconciled with an aspect of the topological fundamental group, so that the "Galois theory of coverings", or at least the part of it relating to finite coverings, can be within the theory rather than just analogous to it;

*the setting becomes rather pure category theory, with the emphasis on the tensor product of étale k-algebras, fibre functors on this category of algebras, and a Galois group as an automorphism group of such a functor.

So the second question is probably: sources of this point of view. (I have to say parenthetically that expositions of Grothendieck are much needed: the attitude that all you need is to read the man himself plus his students isn't unintelligent, but does also seem a sacred cow.) There is much else in SGA1, of obvious fertility.

Third point: I find this in an abstract of a recent paper: "Grothendieck conceived Galois theory as the axiomatic characterization of the classifying topos of a progroup in terms of a representation theorem for pointed Galois Topoi." In other words, the abstract category theory talks about G-sets as a category. Where has this led?

That would be: (Q1) sociological - Galois theory as understood by groups of mathematicians now; (Q2) historical - sorting out SGA1 in its context; and (Q3) surveying further research.

This all connects to several points on the previous question, namely Riemann surface theory as seminal; Galois theory for infinite extensions as a conceptual advance rather than a technical fix; and tensor products of fields (a rather under-exposed piece of algebra). I'd like help with a timeline that brings this all more into focus. If necessary, the "algebraic fundamental group" should appear. Differential Galois theory needs to be fitted in, also, given that it is related to tannakian categories (an observation of slightly obscure origin, possibly Kuga's: his book Galois' dream: group theory and differential equations also has elements of the picture).


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