The Galois Theorem for Rings

The “Spectrum functor” of a ring and its right adjoint are the key to generalizing to the case of rings the notion of split algebra encountered in the case of fields. The same functors make it possible to define the profinite Galois groupoid of a Galois extension of rings. The Galois theorem for rings then exhibits an equivalence between the category of split algebras and that of profinite presheaves on the profinite Galois groupoid. In the case of fields, this reduces to the classical profinite Galois group and the Grothendieck Galois theorem for arbitrary Galois extensions of fields.