Let $K \subset L$ be a finite field extension and $f(x) \in K[x]$ a monic and irreducible polynomial.
Also, $\gcd(\deg f(x),[L:K])=1$.
How to prove that $f(x)$ is irreducible in $L[x]$?
My idea was to show that $L \subset \overline{K}$.
Then I want to consider $K \subset L \subset L(a)$ and $K \subset K(a) \subset L(a)$ to find a root $a \in \overline{K}$ of $f(x)$ to show that $f(x)$ is the minimal polynomial of $a$ over $L$.
But I have some trouble to show it.
I tried this:
Since $K \subset L$ is finite, it implies that $[L:K] < \infty$.
So $L \subset \overline{K}=\lbrace a \in L | a$ is algebraic over $K \rbrace$, as all polynomials of $L$ have roots in $K$.
I'm not sure if this is right. And how to continue?