TITLE: Minimal number of generators ideal in $\mathbb{Z}[x]$
QUESTION [6 upvotes]: Let $\mathfrak{a}=(x^n,\ldots,p^{n-1}x,p^n)$ ideal in $\mathbb{Z}[x]$.
  Show that the minimal number of generators of $\mathfrak{a}$ is $n+1$.

My strategy to prove this is to use one of the consequences of Nakayama Lemma; that is, to show that $\bar{x^n},\ldots,\bar{p^{n-1}x},\bar{p^n}$ is basis for the $\mathbb{Z}[x]/(x,p)\mathbb{Z}[x]$-module (vector space) $\mathfrak{a}/(x,p)\mathfrak{a}$. However, I was not able to show linear independence.

REPLY [3 votes]: This is a sound approach!
Let $R=\Bbb Z[x]$ and
$\newcommand{\pp}{\mathfrak{p}}\pp=(x,p)$. Then $\mathfrak{a}=\pp^n$. So all one has to do is to prove that
$\pp^n/\pp^{n+1}$ has dimension $n+1$ over $R/\pp$. This is equivalent to
$|\pp^n/\pp^{n+1}|=p^{n+1}$, and that will follow by induction from
$|R/\pp^n|=p^{n(n-1)/2}$. But the $a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}$
form a system of coset representatives for $\pp^n$ in $R$
where $0\le a_j<p^{n-j}$.