Polynomial ring over integrally closed subring is integrally closed in polynomial ring
Given: A ring , an integrally closed subring .
To prove: is an integrally closed subring of .
Proof: Suppose satisfies a monic polynomial over :
Here, for all math>i</math>. Write , for greater than the maximum of the degrees of the . Note that if and only if , so it suffices to show that .
satisfies the monic polynomial:
The constant term of this, viewed as a polynomial in , is:
Since all the are in , this whole constant term is an element of .
Fill this in later