Integral Extensions, Integrally Closed
Integrally Closed
The integral closure of R in an extension S
is the set of elements of S that are integral over R.
This is a ring by
corollary 4.
The ring R is integrally closed in S
if R contains all the elements in S that are integral over R.
Use corollary 5
to show the integral closure of R in S is integrally closed in S.
If S is not specified and R is an integral domain,
then S is assumed to be the fraction field of R.
A ring is normal if it is integrally closed and noetherian.
Let R be a ufd, with fraction field F,
and let u be the root of a monic polynomial p(x).
Now x-u is a factor of p(x) in the ring of polynomials
with coefficients in F.
Apply gauss' lemma,
and those coefficients actually belong to R,
hence u lies in R after all.
Every ufd is integrally closed.
For example, the integers are closed in the rationals.