What we now have to prove is that: if an overring R* of R is such that R* is an integral domain, R* is integral over R and the quotient field 𝔏* of R* is a finite algebraic extension of 𝔏, then R* is a local ring and a finite R-module.

It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.

We describe the correspondence between subfields of the field P_a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group π₁(X⧵O).

Algebraic extension

Meaning

Noun