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 . The proof parallels the proof of the number field result.
We first show that the trace (http://planetmath.org/Trace2) maps to . Choose and let be the minimal polynomial for over ; assume is of degree . Let the conjugates of in some splitting field be . Then the are all integral over since they satisfy ’s monic polynomial in . Since the coefficients of are polynomials in the , they too are integral over . But the coefficients are in , and is integrally closed (in ), so the coefficients are in . But is just the coefficient of in , and thus . This proves the claim.
Now, choose a basis of . We may assume by multiplying each by an appropriate element of . (To see this, let . Choose such that . Then and thus ). Define a linear map .
is 1-1, since if , then . But , so is identically zero, which cannot be since is separable over (it is a standard result that separability is equivalent to nonvanishing of the trace map; see for example , Chapter 8).
- 1 P. Morandi, Field and Galois Theory, Springer, 2006.
|Title||integral closures in separable extensions are finitely generated|
|Date of creation||2013-03-22 17:02:12|
|Last modified on||2013-03-22 17:02:12|
|Last modified by||rm50 (10146)|