following a sentence from "p-adic Numbers, p-adic Analysis and Zeta-functions" by Neal Koblitz, page 66:
Let $\pi \in K$ where $K$ is an extension field of $\mathbb Q_p$ (the p-adic rationals) and $ord_p(\pi)=\frac 1 e$ where $e$ is the index of ramification.
Then any $x \in K$ can be written uniquely in the form $\pi^mu$ where $|u|_p=1$ and $m \in \mathbb Z$.
Can you please help me understand why that is right?
Is it because of $e$ being the ramification index and so $ord_p(\pi)=min_{x \in K}(ord_p(x))$?
I ended up solving it on my own so I might as well share the solution..
Since $ord_p(\pi)=min_{x \in K}(ord_p(x))$, $\pi$ has the smallest p-adic valuation. hence, we can multiply it by any $u$ (with $|u|_p=1$), or by itself, to get any nonzero element of K.
Edited:
Given $a \in K$.
$ord_p$ function's image is $\frac{1}{e}\mathbb Z$ (prove it). Hence, $ord_p(a)=\frac m e$. So one can write $a=\pi^mu$ where $u=\frac{a}{\pi^m}$ and hence the uniqueness of this form.