proof about irreducible polynomials with complex roots in Q

Question

I need some idea to proof the next statement.

Let $\alpha$ and $\beta$ two complex roots from irreducible polynomials $f(x)$ and $g(x)$ in $\mathbb{Q}$. Let $K=\mathbb{Q(\alpha)}$ and $L=\mathbb{Q(\beta)}$. Show that $f$ is irreducible in $L[x]$ if and only if $g$ is irreducible in $K[x]$.

Answer 1

From $$ [K(\beta):K][K:\Bbb Q]=[\Bbb Q(\alpha,\beta):\Bbb Q]=[L(\alpha):L][L:\Bbb Q],$$ we see that $$[K(\beta):K]=\deg g\iff [L(\alpha):L]=\deg f.$$