Let $\alpha$ be integral over $\mathbb{Z}$. Let $K=\mathbb{Q}(\alpha)$ be the field generated by $\alpha$ and let $\mathcal{O}_K$ be the ring of elements of $K$ that are integral over $\mathbb{Z}$. A classical result says that:
Claim 1: There exists an integer $d\in\mathbb{Z}$ such that $\mathcal{O}_K\subseteq{1\over d}\mathbb{Z}[\alpha]$.
Let $A=\mathbb{Z}[x_1,x_2,\ldots,x_n]$ be the ring of polynomials with integral coefficients, let $\beta$ be integral over $A$ and $B=\mathbb{Q}(x_1,x_2,\ldots,x_n,\beta)$ be the field generated by $A$ and $\beta$. Let $A'$ be the ring of elements of $B$ which are integral over $A$. then we have:
Claim 2: There exists a polynomial $p\in A$ such that $A'\subseteq{1\over p}A[\beta]$.
This can be proved by a straightforward modification of this proof of Claim 1, see Theorem 4.1.1.
My question: is there a reference to an explicit proof of Claim 2?
Eventually I found a reference:
Paulo Ribenboim, Classical theory of algebraic numbers, Springer-Verlag, 2001, 681 pp.
Section 6.1, Statement B: Let $R$ be an integrally closed domain, $F$ its field of quotients, let $K$ be a separable extension of degree $n$ of $F$, and let $A$ be the integral closure of $R$ in $K$. Then there exist free $R$-modules $M$ and $M'$ of rank $n$, such that $M'\subseteq A\subseteq M$. Explicitly, if $K = F(t)$ with $t \in A$, if $d$ is the discriminant of $t$ in $K|F$, then $$M' = R\oplus Rt \oplus\ldots\oplus Rt^{n-1}= R[t],\;\;\;\;\;\;\; M = (1/d)R[t]. $$