Abstract: A classical theorem of Hermite and Joubert asserts that any field extension of degree n = 5 or 6 is generated by an element whose minimal polynomial is of the form
t^n + c_1 t^{n-1} + ... + c_{n-1} t + c_n,
with c_1= c_3 = 0. We show that this theorem fails for n = 3^m or 3^m + 3^l with m > l >= 0 (and more generally, for n= p^m or p^m + p^l with m > l >= 0, if 3 is replaced by another prime p). We also prove similar results for division algebras and use them to study the structure of the universal division algebra UD(n).