Let $F$ be a field, and let $f(x),g(x)\in F[x]$ be irreducible polynomials over $F$ with $\deg{f}=m$ and $\deg{g}=n$ so that $\textrm{gcd}(m,n)=1$. Now, suppose that $\alpha$ is a root of $f(x)$ in some extension field over $F$. Prove that $g(x)$ is irreducible over $F(\alpha)$.
I already know that its various well-known 'direct proof'.
But, i wonder if it can be proved using the 'proof by contradiction'.
Starting the assumption : '$g(x)$ is reducible over $F(\alpha)$'.
However, i can't find where to deduce contradictions.
Can anyone help me? or give some comment. Thank you!