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