The Galois Theorem of Grothendieck

The Grothendieck approach to the Galois theorem extends the classical Galois bijection to an equivalence of categories. If K ⊆ L is a finite-dimensional Galois extension of fields, a finite-dimensional K-algebra A is split by L when each element a ∈ A is a root of a polynomial p(X) ∈ K[X] which factors in L[X] into distinct linear factors. The corresponding Galois theorem exhibits a contravariant equivalence between the category of finite-dimensional K-algebras split by L and the category of finite sets provided with an action of the Galois group Gal[L : K]. This contains the classical Galois theorem. Indeed, via this equivalence, the K-algebras K ⊆ M ⊆ L split by L coincide with the intermediate field extensions and are in bijection with the quotients of Gal[L : K], which is finite and viewed here as acting on itself. It is a classical result of the theory of group actions that these quotients are themselves in bijection with the subgroups of Gal[L : K].