I am interested in the following theorem.
Let $L/K$ be a field extension of degree $n$. Let $P(X)$ be a degree $d$ polynomial that is irreducible over $K$. If $n$ and $d$ are relatively prime, then $P(X)$ is irreducible over $L.$
I found a proof of this using tower law calculations, but I would like to ask if it can be proved in an alternative way using permutations and conjugates? I want a more explicit proof in order to better understand this theorem. Thank you.