On a Theorem of Hermite and Joubert

Abstract

AbstractA 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 λn + c1λn−1 + ··· + cn−1λ + cn with c1 = c3 = 0. We show that this theorem fails for n = 3m or 3m + 3l (and more generally, for n = pm or pm + pl, if 3 is replaced by another prime p), where m > l ≥ 0. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).