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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.

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From Theorem 1.2 of these notes we learn that 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).

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Thank you--that primitive element theorem is a very nice result. And I realized this morning 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 $F_2$ be a field of characteristic 2 and let $K = F_2(t,u)$ where $t$ and $u$ are variables and $L$ be the splitting field of $(x^2 - t)(x^2 - u)$. Then it turns out that $L = F(a, b)$ where $a^2 = t$ and $b^2 = u$, $L$ is purely inseparable over $K$, and the intermediate fields $F_2(a + yb)$ 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 $F_2(t)$ and therefore separable over $K$, so we have a non-separable extension that is not purely inseparable and we have infinitely many intermediate fields.

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