We consider a finite unramified Galois field extension $L/K$ of non-Archimedean local fields with finite residue fields $l / k$. firstly, some notations: denote by $q \in O_K$ a uniformizer of $O_K$ and $\pi \in O_L$ a uniformizer of $O_L$. Then $n= [L:K]=e \cdot f$ with $f= [l:k]$ and $(q)O_L= (\pi)^eO_L$.
The theory tells that $Gal(L/K)=Gal(l/k)$. A proof I found in Fesenko's and Vostokov's 'Local fields and their extensions' works as follows:
We construct a map $Gal(L/K) \to Gal(l/k)$ by $\sigma \mapsto \overline{\sigma}$. is well defined, because $\sigma(O_L)=O_L$ and $\sigma(\pi O_L)=\pi O_L$
prove this map is surjective
use that the extension is unramified: $\vert Gal(L/K)= [L:K]=[l:k]= Gal(l/k)$
Although I understand the proof, one thing I sorely miss: how the lifts of elements of $Gal(l/k)$ explicitly look like?
More precisely, from Galois theory of finite fields we know $Gal(l/k)$ is generated by relative Frobenius $\bar{F}: l \to l, a \mapsto a^d$ with $d = \vert k \vert = p^s $ ($p$ is the characteristic of $k$). As $Gal(L/K)=Gal(l/k)$, the unique preimage ("lift") $F: L \to L$ of the relative Frobenius $\bar{F}$ generates $Gal(L/K)$.
Question: Is there any way to describe explicitly how the lift $F$ looks like, ie how it acts on elements of $L$? Can we find any canonical generators of $L$ on which the action of $F$ can be described? Or is this lift only an "abstract" construction without any explicite description?
My idea was - since $l = k(\zeta)$ for a appropriate root of unity $\zeta$ - to lift this root with Hensel's lemma. but I not see any reason why this lift of the root generates $L$ as $K$-algebra .