Let $K$ be a field and $| |$ be a discrete valuation . Let $\mathcal{O} = \{k\in K\colon |a|\leq 1\}$ and $\mathcal{P} = \{k\in K\colon |a| < 1\}$. We know that in this case, $\mathcal{P}$ is an principal (maximal) ideal of the ring $\mathcal{O}$, say $\mathcal{P} = \pi\mathcal{O}$. In many books about algebraic number theory, it follows that every $k \in K$ can be written as $\pi^{\nu}\epsilon$, where $\epsilon$ is a unit, but I can't see why that happens.
Do you know why? Thanks in advance.