Question about proof in Neukirch's Algebraic Number Theory I was reading Proposition 2.2 in chapter I of Neukirch (page 6 in my edition), which states the following for an extension of rings $A\subseteq B$: 

(2.2) Proposition. Finitely many elements $b_1,\dots, b_n\in B$ are all integral over $A$ if and only if the ring $A[b_1,\dots,b_n]$ viewed as an $A$-module is finitely generated. 

Neukirch begins the proof by showing that if $b\in B$ is integral over $A$ then $A[b]$ is a finitely generated $A$-module. To do this, he notes that $b$ integral means there is some monic $f(x)\in A[x]$ of degree $n\geq 1$ such that $f(b)=0$. The claim is that $\{1,b,\dots,b^{n-1}\}$ form a generating set for $A[b]$. Neukirch proceeds to take a polynomial $g(x)\in A[x]$ (so that $g(b)$ is an arbritary element in $A[b])$ and states that "we may then write
$$
g(x)=q(x)f(x)+r(x)
$$
for some $q(x),r(x)\in A[x]$ with $\deg(r(x))<n$". 
Here is my problem: $A[x]$ is not a Euclidean domain in general. If $A$ is a field then sure, but if $A=\mathbb{Z}$ then $\mathbb{Z}[x]$ is not Euclidean so it would seem this step in the proof is not justified. What am I missing here? 
 A: This has nothing to do with $\mathbf A[x]$  being Euclidean, nor even $A$$ being a domain.
By induction, you can suppose $B=A[b]$ for a single  integral element $b\in B$.
Indeed, if $\;b^n+a_{n-1}b^{n-1}+\dots +a_1b+a_0=0$  is a monic equation for $b$, then $\;b^n\in \langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle$.
We'll prove  $b^m\in \langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle$ for all $m\ge n$.
To set the inductive step, suppose $b^n,\dots,b^m\in  \langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle$ for some $m$. Then
\begin{align}
b^{m+1}&=b\cdot b^m\in b\,\langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle =\langle \mkern1.5mu b,b^2,\dots b^{n-1}, b^n\mkern 1.5mu\rangle =\langle \mkern1.5mu b,b^2,\dots b^{n-1}\mkern 1.5mu\rangle+\langle \mkern1.5mu b^n\mkern 1.5mu\rangle \\
&\subseteq\langle \mkern1.5mu b,b^2,\dots b^{n-1}\mkern 1.5mu\rangle+\langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle =\langle \mkern1.5mu1,b,\dots b^{n-1}\mkern 1.5mu\rangle.
\end{align}
