This is a problem in James Milne's text on Galois Theory:
Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F] = m$. If $\gcd(m,n)= 1$, show that $f$ is irreducible over $E$.
I have a proof which I think contains a mistake since it appears to work under the weaker condition that $n\nmid m$, but I am unable to locate the error:
Suppose $f$ has a root $\alpha$ in $E$. Since $f$ is irreducible in $F$, we have that $[F(\alpha) : F] = \deg f = n$. Since $E$ contains both $\alpha$ and $F$, $F(\alpha)$ is a subfield of $E$. Now, $$m = [E:F] = [E:F(\alpha)][F(\alpha):F]= n[E:F(\alpha)],$$ contradicting that $\gcd(m,n)=1$.