The Hermite-Joubert problem over p-closed fields

An 1861 theorem of Ch. Hermite asserts that every field extension (and more generally, every etale algebra) E/F of degree 5 can be generated by an element a is an element of E whose minimal polynomial is of the form f(x) = x(5) + b(2)x(3) + b(4)x + b(5). Equivalently, tr(E/F)(a) = tr(E/F)(a(3)) = 0. A similar result for etale algebras of degree 6 was proved by P. Joubert in 1867. It is natural to ask whether or not these classical theorems extend to kale algebras of degree n >= 7. Prior work of the second author shows that the answer is "no" if n = 3(a) or n = 3(a) + 3(b), where a > b >= 0. In this paper we consider a variant of this question where F is required to be a p-closed field. More generally, we give a necessary and sufficient condition for an integer n, a field F-0 and a prime p to have the following property: Every etale algebra E/F of degree n, where F is a p-closed field containing F-0, has an element 0 not equal a is an element of E such that F[a] = E and tr(a) = tr(a(p)) = 0. As a corollary (for p = 3), we produce infinitely many new values of n, such that the classical theorems of Hermite and Joubert do not extend to etale algebras of degree n. The smallest of these new values are n = 13, 31, 37, and 39.