Given a field extension $L/K$, $\alpha, \beta \in L$ and $f,g \in K[x]$ irreducible polynomials with $f(\alpha)=g(\beta)=0$. Then
$ \operatorname{dim}_K(K(\alpha,\beta)) = \deg(f) \cdot \operatorname{dim}_{K(\alpha)}(K(\alpha,\beta)) = \deg(g) \cdot \operatorname{dim}_{K(\beta)}(K(\alpha,\beta))$
holds. I have no idea how to prove that as the degree of the polynomial because the dimension of the splitting field has to only a divisor of the degree but not the exact product.
$f \in K(\beta)[x]$ is ireducible iff $g \in K(\beta)[x]$ is irreducible. Any ideas how to prove that?