TITLE: Forcing as a new chapter of Galois Theory?
QUESTION [61 upvotes]: 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.

REPLY [18 votes]: It seems to me that the groups involved in producing models without AC (whether as groups of permutations of atoms or as groups of automorphisms of complete Boolean algebras) and the forcing constructions themselves are two rather separate things.  The most frequent use of forcing is to produce models of ZFC (not just ZF), and then groups are not involved.  On the other hand, the Fraenkel-Mostowski-Specker method of permutation models for the negation of AC  involves only permutation groups (and normal filters of subgroups), not forcing.  Groups and forcing come together in Cohen's method of symmetric models.  In the original (and still most common) presentation, one has a group acting on the forcing notion.  But even here, the groups and the forcing are less entangled than they seem.  Vopenka and Hajek showed (in their book "Theory of Semisets") how to get Cohen-style models by (1) starting with a permutation model, (2) forcing over it, and (3) passing to the pure (or well-founded) part.  In this presentation, the groups (and filters of subgroups) are only in step (1), and the forcing is only in step (2).  
It seems to me that, if one wants to analyze forcing from an algebraic point of view, one should begin with the simplest case, where no symmetries are involved.  The algebraic side of this is the study of complete Boolean algebras.  Afterward, one can embellish the picture by adding group actions, either acting on the Boolean algebras or producing permutation models over which to force.  The Grigorieff paper that Francois cited is an excellent place to start.  
Though the OP asked about forcing, let me also mention that the non-forcing context of permutation models might be a better place to look for Galois-like phenomena.  To begin, note that a normal filter $\mathcal F$ of subgroups of a group $G$ makes $G$ a topological group, in which $\mathcal F$ generates the neighborhood filter at the identity.  The notion of symmetry with respect to $\mathcal F$ that is used in defining the permutation model is just continuity of the standard action of $G$ on sets.