Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set of units in $\mathcal{O}_K^{\times} $ whose reduction modulo $\{\overline{t_{\alpha}}\} \in k_K$ form a $p$-basis of $k_K$. Generalizing Fontaine-Wintenberger for perfect field case, Scholl in this paper in section 2.3, showed the existence of a field of norm for a Kummer extension $K_\infty$ over $K$. The field of norm is given by $E_K\cong k_K((X))$, (here $X = \bar{\pi}$ in the notation of Scholl)
I am interested in knowing what are the finite separable extensions of this field of norm $E_K$. Is there any closed form for writing the finite separable extensions of $E_K$?
I know that in the perfect case, the field of norm is $F_p((X))$ where $X$ is a uniformizer and finite separable extensions of $F_p((X))$ are of form $F_q((X^{\prime}))$ where $X^{\prime}$ is some other uniformizer and $q$ is a power of $p$.
Is there such a thing in the imperfect case?
(As $k_K$ is imperfect, I know that by primitive element theorem, finite separable extensions of $E_K$ are of form $E_K(\delta)$ for some $\delta$)
Is it true that we can write any finite separable extension of $E_K$ in the form $k_L((X^{\prime}))$ where $X^{\prime}$ is some uniformizer and $k_L$ is a finite separable extension of $k_K$?
Scholl mentions that $k_K$ is formally etale over $F_p(\overline{t_1},...\overline{t_d})$. Does this help to write the finite separable extensions of $E_K$ in terms of $F_p$ and some variables?
