I saw from here that if $K/k$ is a finite field extension, then the following are equivalent:
(1) The extension contains a primitive element (simple extension).
(2) The number of intermediate fields between $k$ and $K$ is finite
The author of the article didn't give the source or proof of this proposition and I wonder if this is true (either direction).
This is true. Here's a sketch of the proof. Assume $k$ is infinite otherwise it's trivially true.
Suppose $K=k(\alpha)$ . Given an intermediate field $L$, look at $min.poly_L(\alpha)$. This is an injection from the set of intermediate fields to the set of factors of $min.poly_k(\alpha)$ in $\bar k[X]$ .The latter is a finite set and hence number of intermediate fields is finite.
Now assume the number of intermediate fields is finite. By induction it suffices to prove the claim for $K=k(\alpha_1,\alpha_2)$. Consider $k(\alpha_1 + t \alpha_2) \ , t\in k$. All these fields can't be distinct and hence $\exists s\neq t \in k$ such that $k(\alpha_1 + s \alpha_2) = k(\alpha_1 + t \alpha_2) $. It is easy to see that both $\alpha_1,\alpha_2$ belong to this simple extension of $k$