abstract algebra - Proof of a Lemma for a local field extension with certain properties

The following result is from "Euler Factors determine local Weil Representations" by Tim and Vladimir Dokchitser:

Lemma 1: Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. Then there exists a cyclic totally ramified extension $L/K$ of degree $e$ such that $FL/L$ is unramified of degree $n$.

The authors argue that it is enough to prove this Lemma for the case of $F/K$ having prime power degree and now I am trying to understand that.

I was able to prove the following result which should be helpful:

Lemma 2: Let $F/K$ be a finite extension with degree $n$ and let $p_1^{n_1} \cdots p_r^{n_r}$ be the prime decomposition of $n$. Then there exist finite extensions $F_i/K$ with $[F_i:K]=p_i^{n_i}$ such that $F$ is the compositum of $F_1,\dots,F_r$.

For simplicity, let $n = p_1^{n_1} p_2^{n_2}$ be the prime decomposition of $F/K$ and let $F = F_1 F_2$ be the corresponding decomposition of $F$ into prime power degree extensions of $K$ obtained from Lemma 2.

If Lemma 1 is true for the case of prime power degree extensions, there exist cyclic and totally ramified extensions $L_1$ and $L_2$ of $K$ of degree $e(F_1/K)$ and $e(F_2/K)$ such that $F_1L_1/L_1$ and $F_2 L_2/L_2$ are totally ramified of degree $p_1^{n_1}$ and $p_2^{n_2}$, resprectively.

To show the general case, I assume that $L$ must be $L_1 L_2$. Then it is unproblematic to show that $L/K$ is unramified of degree $n$ due to the coprimeness of $p_1$ and $p_2$. But I have no idea how to show that $FL/L$ is unramified of degree $n$. It is especially hard because the ground field $L$ is a compositum itself and there are no general rules like $[FL:K] \leq [F:K] [L:K]$ which I could apply here.

Could you please help me completing this proof? Any help is really appreciated!