Let $K\subset K(\alpha)\subset K(\alpha,\beta)$ be field extensions. Then the question is whether the minimal polynomials of $\beta$ over $K$ and $\beta$ over $K(\alpha)$ have degrees where one divides the other; in other words, is it always true that
$$f_{K(\alpha)}^{\beta}\big| f_{K}^{\beta}\implies \deg\left(f_{K(\alpha)}^{\beta}\right)| \deg\left(f_{K}^{\beta}\right)?$$
I'm having trouble finding a counter-example (if it exists). Thanks for any suggestions or hints!
I have found something weird. I have proved the following: $$ \gcd\bigg([K(\alpha):K],[K(\beta):K]\bigg)=1 \implies [K(\alpha,\beta):K]=[K(\alpha):K][K(\beta):K].$$ Now suppose that $\gcd\bigg([K(\alpha):K],[K(\beta):K]\bigg)=1$ which means that: $K(\alpha)\cap K(\beta)=K$, right?
W.l.o.g. assume that $[K(\alpha):K]<[K(\beta):K]$.
By the tower rule and the definition of minimal polynomials, we have $$[K(\alpha,\beta):K]=[K(\alpha,\beta):K(\alpha)]\cdot[K(\alpha):K]=\deg\left(f_{K(\alpha)}^{\beta}\right)\cdot\deg\left(f_{K}^{\alpha}\right).$$ By the theorem above, we get $$[K(\alpha,\beta):K]=[K(\alpha):K]\cdot[K(\beta):K]=\deg\left(f_{K}^{\alpha}\right)\cdot\deg\left(f_{K}^{\beta}\right).$$ Now we have $\deg\left(f_{K(\alpha)}^{\beta}\right)=\deg\left(f_{K}^{\beta}\right)$ as expected. So at least there is a certain condition for which the titled problem is true. How about the inequality where one of these degrees divides the other degree? Could someone help me with that?