Ring of integer-valued polynomials over normal domain is normal

Statement
Suppose $$R$$ is a fact about::normal domain. Let $$\operatorname{Int}(R)$$ denote the fact about::ring of integer-valued polynomials over $$R$$. Then, $$\operatorname{Int}(R)$$ is also a normal domain.

Facts used

 * 1) uses::polynomial ring over a field is normal

Proof
Given: A normal domain $$R$$. $$\operatorname{Int}(R)$$ is the ring of integer-valued polynomials over $$R$$.

To prove: $$\operatorname{Int}(R)$$ is also a normal domain.

Proof: Let $$K$$ be the field of fractions of $$R$$. Now, $$\operatorname{Int}(R)$$ is a subring of $$K[x]$$. We thus have:

$$\operatorname{Int}(R) \subseteq K[x] \subseteq K(x)$$

where $$K(x)$$, the field of rational functions, is the field of fractions of both $$\operatorname{Int}(R)$$ and $$K[x]$$. By fact (1), $$K[x]$$ is normal, so the integral closure of $$\operatorname{Int}(R)$$ in $$K(x)$$ is contained in $$K[x]$$. Thus it suffices to show that elements of $$K[x]$$ that are integral over $$\operatorname{Int}(R)$$ are in $$\operatorname{Int}(R)$$.

Suppose now that $$f \in K[x]$$ is integral over $$\operatorname{Int}(R)$$. In other words, we have a monic polynomial that $$f$$ satisfies with coefficients in $$\operatorname{Int}(R)$$:

$$f^n + g_{n-1}f^{n-1} + \dots + g_0 = 0$$,

where $$g_i \in \operatorname{Int}(R)$$.

Our goal is to show that $$f(a) \in R$$ for any $$a \in R$$.

For any $$a \in R$$, evaluation at $$a$$ gives:

$$(f(a))^n + g_{n-1}(a)f(a)^{n-1} + \dots + g_0(a) = 0$$.

Since all the $$g_i$$ are in $$\operatorname{Int}(R)$$, this is a monic polynomial with coefficients in $$R$$. Further, since $$f \in K[x]$$, $$f(a) \in K$$ satisfies a monic polynomial with coefficients in $$R$$. This forces $$f(a) \in R$$ since $$R$$ is normal.