Let $K=\mathbb{Q}_p$ and $L/K$ be a cyclic and totally ramified extension of degree $n$, generated by an element $\alpha$ (i.e. $L=K(\alpha)$). Let $L'/K$ be another cyclic and totally ramified extension of degree $n$ with $L \neq L'$. Consider the unramified extension $F/K$ of degree $n$ and embed $L,L',F$ in a common field $E$ (for instance, $E = \bar{K}$, an algebraic closure of $K$, or $E = LL' F$, the compositum of $L,L',F$).
Question: Does there exist an element $c \in F^* \subseteq E^*$ such that $L'/K$ is generated by $c \alpha$ (i.e. $L'=K(c \alpha)$)?
I tried to consider the example $L' = (LF)^{\langle \sigma \tau \rangle}$ where $\sigma$ is a generator of $\operatorname{Gal}(LF/L)$, i.e. $L = (LF)^{\langle \sigma \rangle}$, and $\tau$ is a generator of $\operatorname{Gal}(LF/F)$, i.e. $F = (LF)^{\langle \tau \rangle}$. By definition, $\sigma(\alpha) = \alpha$. If $\beta$ is a generator of $F/K$, then $\tau(\beta) = \beta$.
Now it would be interesting to see if one could describe an element $c \in F$ in terms of $\beta$ such that $(\sigma \tau)( c \alpha) = c \alpha$, and I expect this element to be a generator of $L'/K$.
If we have Kummer extensions, i.e. $K$ contains a primitive $n$-th root of unity $\zeta_n$, and $\tau(\alpha) = \zeta_n \alpha$ and $\sigma(c) = \zeta_n^{-1} c$ for an appropriate generator $c$ of $F/K$ (explicitly: $c,\dots,c^{n-1} \not\in K$ and $c^n \in K$), then $(\sigma \tau)(c \alpha) = c \alpha$ indeed.
But I have trouble showing the existence of such an element $c$ for the general case.
Could you please help me with my question and my approach?