Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, why $p＝π^eu$(u:unit element of ring of integers of $L$)?

Question

Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, why $p＝π^eu$ ($u$: unit element of ring of integers of $L$)? From the definition of ramification index,

$(p)＝(π)^e$

Thus, $(p)＝(π^e)$

From here, why can we say $p＝π^eu$ ($u$:unit element of ring of integers of $L$)?