The Hermite-Joubert problem over p-closed fields

An 1861 theorem of Ch. Hermite [He] asserts that every field extension (and more generally, every étale algebra) E/F of degree 5 can be generated by an element a ∈ E whose minimal polynomial is of the form f(x) = x + b2x 3 + b4x+ b5 . Equivalently, trE/F (a) = trE/F (a ) = 0. A similar result for étale algebras of degree 6 was proved by P. Joubert in 1867; see [Jo]. It is natural to ask whether or not these classical theorems extend to étale algebras of degree n > 7. Prior work of the second author shows that the answer is “no” if n = 3 or n = 3 + 3, 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 F0 and a prime p to have the following property: Every étale algebra E/F of degree n, where F is a p-closed field containing F0, has an element 0 6= a ∈ E such that F [a] = E and tr(a) = tr(a) = 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 étale algebras of degree n. The smallest of these new values are n = 13, 31, 37, and 39.