$f,g$ be irr poly of degree $m$ and $n$. Show that if $\alpha$ is a root of $f$ in some extension of $F$, then $g$ is ireducible in $F(\alpha)[x]$

Let $\beta$ be a root of $g$. It is not necessarily the case that $F(\beta)$ is contained in $F(\alpha)$, so we should instead look at the extension $F(\alpha, \beta)$ of $F$, which contains both $F(\alpha)$ and $F(\beta)$. We can write the degree of $F(\alpha,\beta)$ over $F$ in two ways:

$$ [F(\alpha,\beta) : F(\alpha)][F(\alpha) : F] = [F(\alpha, \beta) : F(\beta)][F(\beta) : F]$$

or in other words,

$$ m[F(\alpha,\beta) : F(\alpha)] = n[F(\alpha,\beta) : F(\beta)]$$

Clearly, $n$ divides the product of $m$ (to which it is relatively prime) and $[F(\alpha,\beta) : F(\alpha)]$. If $g$, whose degree is $n$, is not irreducible over $F(\alpha)$, then the extension $[F(\alpha,\beta) : F(\alpha)]$ must be an integer which is strictly less than $n$. See if you can get a contradiction from here.