If $E$ is an extension field of $F$, $\alpha \in E$ is algebraic over $F$, and $\beta \in F(\alpha)$, then deg($\beta, F$) divides deg($\alpha, F$).
I know that deg($\alpha$, $F$) is the degree of the irreducible polynomial for $\alpha$ over $F$ (and similarly for $\beta$). But what does it mean for $\beta$ to be in $F(\alpha$), and what exactly is this theorem saying? Thank you.
This is because $\deg_F\alpha=[F(\alpha):F]$, $\deg_F\beta=[F(\beta):F]$ and $$[F(\alpha):F]=[F(\alpha):F(\beta)]\cdot[F(\beta):F]$$