Manjul Bhargava and Matt Satriano (starting a postdoc at Michigan this fall) posted a nice paper on the arXIv, “On a notion of “Galois closure” for extensions of rings.” The motivation for this work (I’m guessing) comes from Bhargava’s work on parametrizations of number fields. Bhargava needs to generalize many classical objects of algebraic number theory from the setting of fields to rings. For instance, the sextic resolvent of a degree 5 polynomial f(x) (developed by Lagrange, Malfatti, and Vandermonde in the late 18th century) is a degree 6 polynomial g(x) which has a rational root if and only if the equation f(x)=0 is solvable in radicals.
In modern Galois-theoretic language, we would describe the sextic resolvent as follows. If K/Q is the Galois closure of the field generated by a root of f, then the Galois group Gal(K/Q) acts on the 5 roots of f, and is thus identified with a subgroup of S_5. On the other hand, S_5 has an index-6 subgroup H, the normalizer of a 5-cycle. So the action of Gal(K/Q) on S_5 / H gives rise to an extension of Q of degree 6 (maybe a field, maybe a product of fields) and this is Q[x] / g(x).
So far, so good. But Bhargava needs to define a sextic resolvent which is a rank-6 free Z-module, not a 6-dimensional Q-vector space. The sticking point is the notion of “Galois closure.” What is the Galois closure of a rank-5 algebra over Z? Or, for that matter, over a general commutative ring?
Bhargava gets around this question in his paper on quintic fields by using a concrete construction particular to the case of quintics. But in the new paper, he and Satriano propose a very nice (“very nice” means “functorial”) completely general construction of a “Galois closure” G(A/B) for any extension of rings A/B such that A is a locally free B-module of rank n. G(A/B) is a locally free B-module, endowed with an S_n-action, as you might want, and it agrees with the usual definition when A/B is an extension of fields.
But there are surprises — for instance, the Galois closure of the rank 4 algebra C[x,y,z]/(x,y,z)^2 over C is 32-dimensional! In fact, the authors show that there is no definition of Galois closure which is functorial and for which G(A/B) always has the “expected” rank n! over B. This might explain why no one has written down this definition before, and I think it is what gives the paper a sort of offbeat charm. It illustrates a useful point: you’ve got to know when it’s time to mulch an axiom.