abstract algebra - Some field extensions with coprime degrees

Let $L/K$ be a finite field extension with degree $m$. And let $n\in \mathbb N$ such that $m$ and $n$ are coprime. Show the following:

If there is a $a\in \mathbb K$ such that an $n$-th root of $a$ lies in $L$ then we have already $a\in \mathbb K$.

My attempt:

The field extension $K(\sqrt[n]{a})/K$ has degree smaller $n$. The minimal polynomial of
$\sqrt[n]{a}$ namely $m_{\sqrt[n]{a}}(X)\in K[X]$ divides $X^n-a$. I.e. let $k$ be the degree of the minimal polynomial, then $k|n$.

But because of the formula $[L:K]=[L:K(\sqrt[n]{a}]\cdot [K(\sqrt[n]{a}):K]$ $k|m$, hence $k=1$ and hence our conclusion follows because $[K(\sqrt[n]{a}):K]=1$ yields $\sqrt[n]{a}\in K$ .

Can someone go through it? Thanks.