Is it true that $k[\alpha,\beta]=k(\alpha,\beta)$, $\alpha$ and $\beta$ are algebraic over $k$.

Question

Is there a generalization to this question?

Namely, if $k \subset K$ are fields and $\alpha, \beta \in K$ are algebraic over $k$, is it true that $k[\alpha,\beta]=k(\alpha,\beta)$?