an exercise about finite extension field and root of a polynomial

Question

Suppose $K|F$ is a field extension of degree $n$ and $f(X)\in F[X]$ is an irreducible polynomial of degree $m\ge 2$ and $(m,n)=1$. Prove that $f(X)$ has no root in $K$.

thanks

Answer 1

If $f$ has a root $\alpha \in K$, then this gives us $[F(\alpha):F]=m$. But this divides $[K:F]=n$, a contradiction.

(The assumption that $n,m$ are coprime is too strong; we just need that $m$ does not divide $n$.)