# Polynomial ring over integrally closed subring is integrally closed in polynomial ring

From Commalg

## Statement

Suppose is an integrally closed subring of a commutative unital ring . Then, the polynomial ring is an integrally closed subring of .

## Proof

**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*