TITLE: Finite extension of fields with no primitive element
QUESTION [34 upvotes]: What is an example of a finite field extension which is not generated by a single element?
Background: A finite field extension E of F is generated by a primitive element if and only if there are a finite number of intermediate extensions. See, for example, [Lang's Algebra, chapter V, Theorem 4.6].

REPLY [2 votes]: If you're willing to work with commutative rings instead of just fields, you can gain some Galois-theoretic insight by base-changing to an algebraic closure $\bar{K}$ (or even a suitably large finite field extension).  Tensor product takes any primitive extension of your original field $K$ to a $\bar{K}$-algebra generated by a single element, so it suffices to find an extension that base changes to a $\bar{K}$-algebra generated by at least 2 elements.
If we take Alon Amit's minimal example $K = F(x,y), L = F(x^{1/p},y^{1/p})$, we can tensor it up to $\bar{K}[r,s]/(r^p-x,s^p-y)$.  By a suitable change of coordinates, we see that the spectrum of $\bar{K} \otimes_K L$ is a trivial torsor under $\alpha_p \times \alpha_p$ (and $\mu_p \times \mu_p$ and various other groups).  This description descends to $L$, but the torsor becomes nontrivial over $K$.  However, we do see that there is a 2-dimensional tangent space, and this gives us the derivations in KConrad's answer.  Furthermore, intermediate fields are fixed by subgroup schemes, and since we are in the the height one setting, these are cut out by the infinitely many $K$-rational slopes in the tangent space.
It's been almost 5 years since JBorger's comment to KConrad's answer, so I'm not sure if he's still interested.  Anyway, there is a Galois theory with group schemes that can be applied to purely inseparable extensions, worked out more or less in Chase, S.U.: Infinitesimal Group Scheme Actions on Finite Field Extensions Am. J. Math., Vol. 98, No.2, pp. 441-480 (1976).  Basically, you can corepresent the functor $A \mapsto \text{Aut}_A(L \otimes_K A)$ by a Hopf algebra over $K$, whose spectrum is called the Galois group scheme of $L/K$.  As far as I can tell, in our case, it is the Weil restriction of $\alpha_p \times \alpha_p$ (or $\mu_p \times \mu_p$, etc. - these are not canonical) along $L/K$, and the coordinate ring has dimension $(p^2)^{p^2}$.  I think there is a Galois correspondence involving subgroup schemes and descent data, but I haven't actually read the paper thanks to the paywall.