Abstract view
On a Theorem of Hermite and Joubert


Published:19990201
Printed: Feb 1999
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 $\lambda^n + c_1 \lambda^{n1} +
\cdots + c_{n1} \lambda + c_n$ with $c_1=c_3=0$. We show that this
theorem fails for $n=3^m$ or $3^m + 3^l$ (and more generally, for $n =
p^m$ or $p^m + p^l$, if 3 is replaced by another prime $p$), where $m
> l \geq 0$. 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 $\UD(n)$.