On page 68 of the book p-adic Numbers, p-adic Analysis, and Zeta-Functions by Koblitz, the following corollary is stated:
If $K$ is a finite extension of $\mathbb{Q}_p$ of degree $n$, index of ramification $e$, and residue field degree $f$, and if $\pi$ chosen so that $\text{ord}_p \pi = \frac{1}{e}$, then every $\alpha\in K$ can be written in one and only one way as $$\sum_{i=m}^\infty a_i\pi^i$$where $m=e\cdot \text{ord}_p\alpha$ and each $a_i$ satisfies $a_i^{p^f}=a_i$
The proof of this corollary has been left as an exercise to the reader (me), with the remark that it is easy. However, I have no idea where to start. The proposition before this corollary is
There is exactly one unramified extension $K^{\text{unram}}_f$ of $\mathbb{Q}_p$ of degree $f$, and it can be obtained by adjoining a primitive $(p^f - 1 )$th root of $1$. If $K$ is an extension $\mathbb{Q}_p$ of degree $n$, index of ramification $e$, and residue field degree $f$ (so that $n = ef$), then $K = K^{\text{unram}}_f(\pi)$, where $\pi$ satisfies an Eisenstein polynomial with coefficients in $K^{\text{unram}}_f$.
Any help would be greatly appreciated.