A sufficient condition for irreducibility of a polynomial in an extension field.

Question

Here is another problem of the book Fields and Galois Theory by Patrick Morandi, page 37.

Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$. If $\gcd(n,m)=1$, show that $f$ is irreducible over $K$.

I'm preparing for my midterm exam so I'm trying to solve as many as this book problems. Thanks for your helps.

Answer 1

Let $L \supset K$ be obtained by adjoining a root of $f$ to $K$, say $\alpha$. Then consider the chains $F \subset K \subset L$ and $F \subset F(\alpha) \subset L$. You know $[F(\alpha):F] = n$, because $f$ is irreducible over $F$. What does this tell you about $[L:K]$?