integral closures in separable extensions are finitely generated
The theorem below generalizes to arbitrary integral ring extensions (under certain conditions) the fact that the ring of integers
of a number field
is finitely generated
over ℤ. The proof parallels the proof of the number field result.
Theorem 1.
Let B be an integrally closed Noetherian domain with field of fractions
K. Let L be a finite separable extension
of K, and let A be the integral closure of B in L. Then A is a finitely generated B-module.
Proof.
We first show that the trace (http://planetmath.org/Trace2) TrLK maps A to B. Choose u∈A and let f=Irr(u,K)∈K[x] be the minimal polynomial for u over K; assume f is of degree d. Let the conjugates of u in some splitting field
be u=a1,…,ad. Then the ai are all integral over B since they satisfy u’s monic polynomial
in B[x]. Since the coefficients
of F are polynomials
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 TrLK(u) is just the coefficient of xd-1 in f, and thus TrLK(u)∈B. This proves the claim.
Now, choose a basis ω1,…,ωd of L/K. We may assume ωi∈A by multiplying each by an appropriate element of B. (To see this, let Irr(ωi,K)∈K[x]=xd+k1xd-1+…+kd. Choose b∈B such that bki∈B∀i. Then (bω)d+bk1(bω)d-1+…+bdkd=0 and thus bω∈A). Define a linear map φ:L→Kd:a↦(TrLK(aω1),…,TrLK(aωd)).
φ is 1-1, since if u∈kerφ,u≠0, then Tr(uL)=0. But uL=L, so TrLK is identically zero, which cannot be since L is separable over K (it is a standard result that separability is equivalent to nonvanishing of the trace map; see for example [1], Chapter 8).
But TrLK:A→B by the above, so φ:A↪Bd. Since B is Noetherian, any submodule of a finitely generated module is also finitely generated, so A is finitely generated as a B-module.
∎
References
-
1
P. Morandi, Field and Galois Theory
, Springer, 2006.
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 |