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.
Composition
If two extensions of R are integrally closed, their composition need not be.
Adjoin the square root of -1, or the square root of 2, to the integers.
Either extension alone is integrally closed.
Combine the two extensions, having basis 1, i, sqrt(2), and sqrt(2)i, and show that the eighth root of 1 is not spanned.
In other words, no integer combination of 1 and sqrt(2) yields sqrt(1/2).
However, the eighth root of 1, which is half of sqrt(2)+sqrt(2)i, is integral over the integers.