Complete D.V.R's That have different characteristic than the residue field

Question

I'm working through Local Fields by Serre and am stumped by something that he thinks should be obvious.

Let $A$ Be a complete D.V.R with uniformizer $\pi$ and $\overline{K}$ be it's residue field. $A$ has characteristic $0$ and $\overline{K}=p>0$. Let $S$ be a set a representatives of $\overline{K}$ in $A$. I know that $A$ is a free $\mathbb{Z_p}$ -module on the set $\{\pi^{n}\}_{n=0}^{\infty}$. If $e=v(p)$ in $A$. He states that if $|\overline{K}|=p^{f}$, that $A$ is a free module of rank $ef$.

Answer 1

If $K$ is the complete field corresponding to $A$, under your hypotheses $[K:{\mathbb Q}_p]=ef$ and if $\overline{a}_1,\ldots,\overline{a}_f$ is a basis for the residue field $\overline{K}$ over ${\mathbb F}_p$, then the $ef$ elements $a_i\pi^j$, $1\leq i\leq f$, $0\leq j\leq e-1$ are a basis for $A$ over ${\mathbb Z}_p$.