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Introduction

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

xi

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