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There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was talking about: Galois Theory, far from being confined to its primary example, namely field extensions, is pervasive throughout mathematics, and still to be fully understood.

Ever since I first heard of forcing I was struck by its compelling analogy with field extensions: the ground model (read Q), the generic G (read a new element, say $\sqrt{2}$), the new M[G] (read Q[$\sqrt{2}$]), etc.

I quote Joel Hamkins 's words here, in his sparkling paper on the multiverse:

In eﬀect, the forcing extension has adjoined the “ideal” object G to V , in much the same way that one might build a ﬁeld extension such as Q[$\sqrt{2}$]

Of course, matters are a bit more complicated in set theory, you have to make sure the extension satisfies the axioms, that it does not "bother" the ordinals, and so on.

Yet, one cannot really think that the analogy stops here.

And it does not: in Galois theory, the main thing is the central theorem, establishing the Galois Connection between subfields of the extension and the subgroups of the Galois group, ie the group of the automorphisms of the extension leaving fixed the underlying field.

No Galois connection, no Galois Theory. But wait, first we need the group. So where is it?

A hint is in the great classic result by Jech-Sochor on showing the independence of AC: by considering a group of automorphisms of P, the ordered set of forcing conditions, one can obtain a new model which is (essentially) the set of fixed points of the induced automorphisms. This is even clearer when one looks at it from the point of view of boolean valued models: each automorphism of the boolean algebra induces an automorphism of the extended universe.

Now my question: is there some systematic work on classifying forcing extensions by their Galois group? Can one develop a full machinery which will apply to relative extensions?

NOTE: I think this is no idle brooding: someone for instance has asked here on MO about the 2-category of the multiverse. That is a tough question, and my sense is that before giving it a satisfactory answer some preliminary work needs to be done. Which work? Well, one needs to re-think the classical set constructions from a structural standpoint, leaving behind its gory technical details. Now, forcing is a huge part of the multiverse, and understanding the structural algebra underpinning it would be, I trust, a huge step toward an algebraic understanding of the multiverse.

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