Number of intermediate fields in non-separable extensions that are also not purely inseparable

Question

If a field extension is separable then we know there are only finitely many intermediate fields. If a field extension is purely inseparable then it is possible for there to be infinitely many intermediate fields.

My question is what can we say about field extensions that are not separable but also not purely inseparable. Is it possible for there to be infinitely many intermediate fields or can it be proved that there are only finitely many intermediate fields?

A proof or an example would be very helpful here.

Answer 1

A finite field extension has only finitely many intermediate fields if and only if the extension is primitive. Then you can easily find examples of primitive extensions which are neither separable nor purely inseparable (by considering roots of such polynomials).

Answer 2

I realized that there are also examples of extensions that are not separable or purely inseparable with infinitely many intermediate fields, so anything can happen here.

An example is the following: let $\mathbb F_2$ be a field of characteristic 2 and let $K =\mathbb F_2(t,u)$ where $t$ and $u$ are variables and $L$ be the splitting field of $(x^2 - t)(x^2 - u)$. Then it turns out that $L =\mathbb F_2(a, b)$ where $a^2 = t$ and $b^2 = u$, $L$ is purely inseparable over $K$, and the intermediate fields $\mathbb F_2(a + yb)$ are all distinct where $y$ ranges over the distinct elements of $K$. Since $K$ is infinite then there are infinitely many intermediate fields. Now let $M = L(c)$, where $c^3 = t$; then $c$ is separable over $\mathbb F_2(t)$ and therefore separable over $K$, so we have a non-separable extension that is not purely inseparable and we have infinitely many intermediate fields.