integral closures in separable extensions are finitely generated

The theorem below generalizes to arbitrary integral ring extensionsPlanetmathPlanetmath (under certain conditions) the fact that the ring of integersMathworldPlanetmath of a number fieldMathworldPlanetmath is finitely generatedMathworldPlanetmathPlanetmathPlanetmath over . The proof parallels the proof of the number field result.

Theorem 1.

Let B be an integrally closedMathworldPlanetmath Noetherian domain with field of fractionsMathworldPlanetmath K. Let L be a finite separable extensionMathworldPlanetmath of K, and let A be the integral closure of B in L. Then A is a finitely generated B-module.


We first show that the trace ( TrKL maps A to B. Choose uA and let f=Irr(u,K)K[x] be the minimal polynomial for u over K; assume f is of degree d. Let the conjugatesPlanetmathPlanetmathPlanetmath of u in some splitting fieldMathworldPlanetmath be u=a1,,ad. Then the ai are all integral over B since they satisfy u’s monic polynomialMathworldPlanetmath in B[x]. Since the coefficientsMathworldPlanetmath of F are polynomialsMathworldPlanetmathPlanetmath in the ai, they too are integral over B. But the coefficients are in K, and B is integrally closed (in K), so the coefficients are in B. But TrKL(u) is just the coefficient of xd-1 in f, and thus TrKL(u)B. This proves the claim.

Now, choose a basis ω1,,ωd of L/K. We may assume ωiA by multiplying each by an appropriate element of B. (To see this, let Irr(ωi,K)K[x]=xd+k1xd-1++kd. Choose bB such that bkiBi. Then (bω)d+bk1(bω)d-1++bdkd=0 and thus bωA). Define a linear map φ:LKd:a(TrKL(aω1),,TrKL(aωd)).

φ is 1-1, since if ukerφ,u0, then Tr(uL)=0. But uL=L, so TrKL is identically zero, which cannot be since L is separable over K (it is a standard result that separability is equivalentPlanetmathPlanetmath to nonvanishing of the trace map; see for example [1], Chapter 8).

But TrKL:AB by the above, so φ:ABd. Since B is NoetherianPlanetmathPlanetmathPlanetmath, any submodule of a finitely generated module is also finitely generated, so A is finitely generated as a B-module. ∎


Title integral closures in separable extensions are finitely generated
Canonical name IntegralClosuresInSeparableExtensionsAreFinitelyGenerated
Date of creation 2013-03-22 17:02:12
Last modified on 2013-03-22 17:02:12
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 13B21
Classification msc 12F05
Related topic IntegralClosureIsRing