Non-normal and inseparable extension of degree 5

Question

I know that it is possible to have a non-normal and inseparable extension of degree $pq$ for any prime $p$ and any natural number $q \ne 1$. But is there possible to construct such extension of degree $5$?

Answer 1

This is not possible.

Let $L/K$ be an extension of degree $5$. Take $\alpha \in L-K$. Then $K(\alpha)$ is a sub-extension of $L$, and hence $[K(\alpha):K]$ divides $[L:K] = 5$, so that degree is either $1$ or $5$. By choice of $\alpha$, it is not $1$. Therefore, this is a primitive inseparable extension generated by any element not in $K$.

Next, the extension must be purely inseparable. We have $$[L:K] = [L:K]_s [L:K]_i$$ and you want $[L:K]_i > 1$. Just like before, the degree is $5$, a prime, and so together these require $[L:K]_i = 5$.

This means that a primitive element $\alpha$ is purely inseparable of degree over $K$ and so it has minimal polynomial $(x-\alpha)^5$, which splits in $L$. This holds for all $\alpha \in L - K$, and obviously the minimal polynomial of anything in $K$ splits in $L$, so the extension is normal.