Let $K= (K, m)$ be a local field (therefore a complete, non-archimedian field with discrete valuation $v: K \to \mathbb{Z} \cup \{ \infty \}$) with perfect resudue field $k$ of characteristic $p \ge 0$.
Assume $L/K$ is a finite field extension of degree $n=[L:K]$ which is assumed to be tamely ramified (i.e. $e:= e(L/K)$ is prime to characteristic $p$). Additionally assume $K$ contains all $n$-roots $\zeta^i _n$ with $0 \le i < n$.
recall $e$ is defined as unique number $m_L^e = m_K O_L$.
I want to show that $L/K$ is cyclic, i.e. $Gal(L/K) = \mathbb{Z}/n$.
My attempts: I want to use Kummer theory. Thus I have to show that $K$ contains all $n$-th roots (this was the assumption), the extension has the form $L= K(a)$ with $a$ root of minimal polynomial $X^n -c$ with $c \in K $.
does anybody have an idea how to manage it?