proof of finite inseparable extensions of Dedekind domains are Dedekind
Let be a Dedekind domain with field of fractions and be a field extension. We suppose that has characteristic (http://planetmath.org/characteristic) and that there is a such that for all . In particular, this is satisfied if it is a purely inseparable and finite extension.
We show that the integral closure of in is a Dedekind domain.
We cannot apply the same method of proof as for the proof of finite separable extensions of Dedekind domains are Dedekind, because here does not have to be finitely generated as an -module.
We use the characterization of Dedekind domains as integral domains in which all nonzero ideals are invertible (see proof that a domain is Dedekind if its ideals are invertible). Note that for any , is in and is integral over so, by integral closure, .
So, let be a nonzero ideal in , and let be the ideal of generated by terms of the form for ,
Then, as is a Dedekind domain, there is a fractional ideal of such that , and write for the fractional ideal of generated by . Then,
On the other hand, if and then
|Title||proof of finite inseparable extensions of Dedekind domains are Dedekind|
|Date of creation||2013-03-22 18:35:42|
|Last modified on||2013-03-22 18:35:42|
|Last modified by||gel (22282)|