Proving $f\in K(\beta)[X]$ irreducible iff $g\in K(\alpha)[X]$ irreducible

Question

Let $K\subset L$ a field extension and $\alpha,\beta\in L$ with minimal polynomials $f,g\in K[X]$.

How to show that $f\in K(\beta)[X]$ is irreducible iff $g\in K(\alpha)[X]$ is irreducible?

Answer 1

Consider the degree of the extension $K\subset K(\alpha, \beta)$.

Note that "$f\in K(\beta)[X]$ is irreducible" and "$g\in K(\alpha)[X]$ is irreducible" are both equivalent to that degree being $\deg(f)\deg(g)$.