Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
In several algebra books there are chapters devoted to integral ring extensions. (I refer to Lang's Algebra, Chapter VII "Extensions of Rings".) Here there are discribed "integrally closed" rings $A$ and there field of fractions $K$, together with a Galois extension $L$ over $K$ and the integral closure $B$ of $A$ in $L$. Instead of the roots of irreducible polynomials in $K$, that are split in $L$, the automorphisms of the Galois group now permute prime ideals of $B$ lying above a fixed prime ideal in $A$.
These kind of ring extensions have several more interesting properties. For details and the definitions of mentioned notions are given in the cited text.
See the following paper for an accessible introduction, and see also the answers to the MO question Is there a Galois correspondence for ring extensions?
M. Ferrero; A. Paques. Galois Theory of Commutative Rings Revisited.