The Galois Theorems in Arbitrary Dimension

In Chapter 3 we pointed out an equivalent definition of a finite-dimensional Galois extension in terms of a tensor product, instead of polynomials over a field. This is a first step towards a Galois theory for rings, where the polynomial approach fails to work. The present chapter develops a second important step in the same direction: getting rid of the notion of dimension, which does not naturally make sense in the case of rings. We thus generalize both the classical Galois theorem and the Grothendieck Galois theorem, from the finite-dimensional case to the case of a Galois extension of fields K ⊆ L of arbitrary dimension. This requires introducing, on the corresponding Galois group Gal[L : K], a profinite topology, as studied in Chapter 3 . The classical Galois theorem then exhibits a bijection between the intermediate field extensions and the closed subgroups of the Galois group. In the Grothendieck approach in terms of split algebras, the Gal[L : K]-sets are now provided with a profinite topology making the group actions continuous.