field and Galois theory

["George Janelidze" <janelidze@mat.ua.pt>]

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