I have a question about @nguyen quang do's answer given in following thread:
correspondence between totally, tamely ramified extension and value groups
We consider a totally tamely ramified field extension $L/K$ of degree $e $. Why does it already imply that $L=K({\pi}_L)$.
Here $\pi_L$ is the unique maximal/prime ideal of the ring $\mathcal{O}_L$.
Remark: I'm working with the definitions for totally/ tamely Ramifications from here: http://alpha.math.uga.edu/~pete/8410Chapter4.pdf
My considerations:
By definition we have $K = K(\pi_K), L = L(\pi_L)$ and $\pi_K \mathcal{O}_L = \pi_L ^e \mathcal{O}_L$, right?
From here I'm stuck. Does it suffice to show that from $l \in L $ we can conclude that $l^e \in \pi_K \mathcal{O}_L$. If yes, how could we pull here the root back?