proof of finite inseparable extensions of Dedekind domains are Dedekind

Let R be a Dedekind domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K and L/K be a field extension. We suppose that K has characteristicPlanetmathPlanetmath ( p>0 and that there is a q=pr such that xqK for all xL. In particular, this is satisfied if it is a purely inseparable and finite extensionMathworldPlanetmath.

We show that the integral closureMathworldPlanetmath A of R in L 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 A does not have to be finitely generatedMathworldPlanetmathPlanetmath as an R-module.

We use the characterization of Dedekind domains as integral domainsMathworldPlanetmath in which all nonzero ideals are invertiblePlanetmathPlanetmathPlanetmath (see proof that a domain is Dedekind if its ideals are invertible). Note that for any xA, xq is in K and is integral over R so, by integral closure, xqR.

So, let 𝔞 be a nonzero ideal in A, and let 𝔟 be the ideal of R generated by terms of the form aq for a𝔞,


Then, as R is a Dedekind domain, there is a fractional idealMathworldPlanetmath 𝔟-1 of R such that 𝔟𝔟-1=R, and write 𝔟A-1 for the fractional ideal of A generated by 𝔟-1. Then,

1R=𝔟𝔟-1𝔞q𝔟A-1. (1)

On the other hand, if a1,,aq𝔞 and b𝔟-1 then


so a1aqb is integral over R and is in A. Therefore, 𝔞q𝔟A-1A. Combining with (1) gives 𝔞q𝔟A-1=A, so 𝔞 is invertible with inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath 𝔞q-1𝔟A-1.

Title proof of finite inseparable extensions of Dedekind domains are Dedekind
Canonical name ProofOfFiniteInseparableExtensionsOfDedekindDomainsAreDedekind
Date of creation 2013-03-22 18:35:42
Last modified on 2013-03-22 18:35:42
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 13A15
Classification msc 13F05