If $K(a,b)$ is a extension field of the field $K$, where $a$, $b$ are algebraic over $K$ and $b$ is separable. We know that if $K$ is infinite, we can find $u$ of $K$ such that $K(a,b)=K(a+bu)$, so there are only a finite number of intermediate fields between $K$ and $K(a,b)$.
Can you give a proof of this result (there are only a finite number of intermediate fields between $K$ and $K(a,b)$) without the primitive element theorem?