Let $L/K$ be a finite totally ramified extension of a local field $K$. Let $E$ and $E'$ be subextensions in $L/K$, linearly disjoint over $K$ with $E E'=L$.
Is there a method to compute a uniformizer $\pi_L$ in $L$ with the two uniformizers $\pi_E$ and $\pi_{E'}$?
In general there exists a polynomial $g \in K(\pi_{E'})[X]$ satisfying $\pi_L = g(\pi_E)$. Is it possible to get a more specific relation between $\pi_L$, $\pi_E$ and $\pi_{E'}$?
I tried to work by defining $\pi_E:= N_{L/E}(\pi_L)$ and $\pi_{E'}$ resp., but it lead nowhere.