An 1861 theorem of Ch. Hermite [He] asserts that every field extension (and more generally, everyétale algebra) E/F of degree 5 can be generated by an element a ∈ 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 forétale 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 toétale 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: Everyétale algebra E/F of degree n, where F is a p-closed field containing F 0 , has an element 0 = a ∈ 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 toétale algebras of degree n. The smallest of these new values are n = 13, 31, 37, and 39.
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