2014/5/19 17:41
It is not easy to give a de finitio n of Galois theory. T here are those who formu-
late it as an order reversing correspondence between the lattice of subobjects
of a certain object (say, the subfields of a finite normal separable field ex-
tension) and the lattice o f subobjects of a simpler object (say, the subgroups
of a finite group). One way to say this is to s ay that we have the same ab-
stract lattice realized in two different contexts, one simpler than the other.
But then we have to say why one is simpler (not an easy task when we are
talking about, as here, the lattice of certain subgroupoids of a type of profinite
groupoid, the subgroupoids being intersections of op en–closed subgroupoids).
Even in the classic case of finite normal separable field extensions it is some-
how the existence of the correspondence that is the point, notwithstanding
any simplification that may result. In fact the teaching of this Galois theory,
a traditional highpoint of a c ourse in abstract algebra usually is framed in
terms of its projected use as a tool in solving polynomial equations. If that
were truly the case, we would teach the (difficult) task of finding all subgro ups
of a group of automorphisms and the (very difficult) task of finding the fixed
field of a subgroup of the automorphism group in other than the few classic
examples typically presented. (The late Don Knutson once r emarked to me
that Galois Theory isn’t about solving equations; it’s about not solving equa-
tions.) One thing which gets obscured in this classic Galois theory is where the
group itself comes from, in pa rt because the description is so straightforward
(the group of automorphisms of the extension). But actually it is more subtle.
The insight here is due to Grothendieck, who was looking for a nother type
of group in another co ntext. To back up a bit with fields, the Galois theory
of all finite separable extension fields of a given field is in terms of the Galois
group of the separable clo sure of the base field. The connection between finite

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