The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely inseparable too.

I was wondering the following:

Is it possible to construct a finite extension of fields $K/F$ which is not simple and such that the purely inseparable closure of $F$ in $K$ is simple?

The purely inseparable closure of $F$ in $K$, denoted $I$, is defined to be the set of all $\alpha \in K$ that are purely inseparable over $F$. Similarly, the separable closure of $F$ in $K$ is denoted $S$ and is defined to be the set of all $\alpha \in K$ that are separable over $F$. One can show that $I$ and $S$ are subfields of $K$ containing $F$.

Can anyone give me a hint on how to construct, or even search for, such an extension?

I know that we cannot have $SI = K$, because we can show from the hypotheses that $SI$ is a simple extension of $F$. In particular, $K$ cannot be normal over $F$ (see Proposition V.6.11 in Lang's Algebra, page 251, third edition). Moreover, $F$ cannot be perfect because every algebraic extension of a perfect field is separable (see Corollary V.6.12 in Lang's Algebra, page 252, third edition), so in particular $F$ cannot be a finite field or have characteristic $0$. Moreover, this answer by @KConrad on MathOverflow shows that every finite extension of $\Bbb{F}_p(x)$ is simple, so these are also ruled out as candidates for $F$.

I haven't been able to make any progress beyond this, though. I've tried fiddling around with the above standard example of $k(x,y)/k(x^p,y^p)$, but no matter how I make $K/F$ non-simple I end up with a non-simple purely inseparable closure. Maybe it's because the only example I know of a non-simple extension is the one mentioned at the beginning that I don't know how to create new examples.

I spent a bit of time trying to prove that no such extension could exist, but could not make any progress there either. Any help is appreciated.

This question is inspired by the problem asked here: A finite extension is simple iff the purely inseparable closure is simple?. Any such example of extension fields $K/F$ would provide a counter-example to the claim in the title of the linked question.


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