Minimum product of degrees of generators of finite field extension Suppose $L/K$ is a finite extension of fields. When is it true that $$\min_{\substack{\{\alpha_1, \ldots, \alpha_n\} \\ L = K(\alpha_1, \ldots, \alpha_n)}} \left(\prod_{i=1}^n [K(\alpha_i): K] \right) = [L:K]?$$ By the primitive element theorem, this is certainly true if $L/K$ is separable. This paper seems to suggest the condition does not hold in general for purely inseparable extensions. But I cannot find or think of an explicit counterexample; does anybody know of one?
 A: I came across this question with a collaborator while thinking about something similar. I believe the example which follows will work.
For ease of exposition, let $k=\mathbb F_2$, though we explain how to generalise at the end. Let $k_0=k(t,u,v)$, $k_1=k_0(t^{1/2})$ and let $k_2$ be the splitting field of the polynomial $f=(x^2+ut^{1/2})(x^2+(t^{1/2}+v))$. Thus $k_2=k_1(\alpha,\beta)$ where $\alpha=u^{1/2}t^{1/4}$, $\beta=(t^{1/2}+v)^{1/2}$.
Then $k_2$ is a non-simple purely inseparable extension of $k_1$ of degree $4$ and of $k_0$ of degree $8$.
We claim every element of $k_2\setminus k_1$ is of degree $4$ over $k_0$. This then shows that $k_2/k_0$ gives an example of the desired sort, since one needs at least two elements of $k_2$ to generate $k_2$ over $k_1$: the left-hand side of the equation in the question is then $16$ and the right-hand side is $8$.
We have $\{1,\alpha,\beta,\alpha\beta\}$ is a basis of $k_2$ over $k_1$. Let $\gamma=a+b\alpha+c\beta+d\alpha\beta$ for $a,b,c,d\in k_1$. Then $\gamma^2=(a^2+c^2v+d^2ut)+(b^2u + c^2+d^2uv)t^{1/2}$, written in terms of a basis $\{1,t^{1/2}\}$ of $k_1$ over $k$. Suppose $\gamma^2\in k_0$. Then $$b^2u + c^2+d^2uv=0.$$ As $b^2,c^2,d^2\in k(t,u^2,v^2)$, then viewing this as a polynomial in $v$, we must have $d=0$; in turn, viewed as a polynomial in $u$, we have $b=0$, which implies $c=0$ also. But this implies $\gamma\in k_1$ which proves the claim.
More generally, the same idea works with $k$ any field of characteristic $p$, and we let $k_0=k(t,u,v)$, $k_1=k_0(t^{1/p})$, $k_2=k_1(u^{1/p}t^{1/p^2}, (t^{1/p}+v)^{1/p})$.
Edited to add: we discovered that [Richard Rasala, Inseparable Splitting Theory, Transactions of the American Mathematical Society , Dec., 1971, Vol. 162 (Dec., 1971), pp. 411-448] has a very similar example (Example 2, p413) and some other ones as well.
