I have problems with this exercise
Let be $K(\alpha)/K$ and $K(\beta)/K$ disjoint extensions with at least one of them odd degree. Prove that $\alpha\beta$ is a primitive element for the extension $K(\alpha,\beta)/K$.
Some of my ideas were
Prove that $K(\alpha,\beta) \subset K(\alpha\beta)$ or that $K(\alpha) \subset K(\alpha\beta)$.
Use that in this situation $K(\alpha)=K(\alpha^2)$.
Tried to relate the irreducible polynomials from the extensions involved.
I didn't find anything useful. Can you help me?
Thank you in advance.