proof of finite separable extensions of Dedekind domains are Dedekind
Let be a Dedekind domain with field of fractions and be a finite (http://planetmath.org/FiniteExtension) separable extension of fields. We show that the integral closure of in is also a Dedekind domain. That is, is Noetherian (http://planetmath.org/Noetherian), integrally closed and every nonzero prime ideal is maximal (http://planetmath.org/MaximalIdeal).
First, as integral closures are themselves integrally closed, is integrally closed. Second, as integral closures in separable extensions are finitely generated, is finitely generated as an -module. Then, any ideal of is a submodule of , so is finitely generated as an -module and therefore as an -module. So, is Noetherian.
It only remains to show that a nonzero prime ideal of is maximal. Choosing any there is a nonzero polynomial
for , and such that . Then
so is a nonzero prime ideal in and is therefore a maximal ideal. So,
|Title||proof of finite separable extensions of Dedekind domains are Dedekind|
|Date of creation||2013-03-22 18:35:36|
|Last modified on||2013-03-22 18:35:36|
|Last modified by||gel (22282)|