1. CJM 2007 (vol 59 pp. 658)
 Mináč, J.; Wadsworth, A.

Division Algebras of Prime Degree and Maximal Galois $p$Extensions
Let $p$ be an odd prime number, and let $F$
be a field of characteristic not $p$ and not containing
the group $\mu_p$ of $p$th roots of unity.
We consider cyclic $p$algebras over $F$ by descent from
$L = F(\mu_p)$. We generalize a theorem of Albert by
showing that if $\mu_{p^n} \subseteq L$, then a division
algebra $D$ of degree $p^n$ over $F$ is a cyclic
algebra if and only if there is $d\in D$ with $d^{p^n}\in
F  F^p$. Let $F(p)$ be the maximal $p$extension
of $F$. We show that $F(p)$ has a noncyclic algebra
of degree $p$ if and only if a certain eigencomponent of the
$p$torsion of $\Br(F(p)(\mu_p))$ is nontrivial.
To get a better understanding of $F(p)$, we consider
the valuations on $F(p)$ with residue characteristic
not $p$, and determine what residue fields and value
groups can occur. Our results support the conjecture
that the $p$ torsion in $\Br(F(p))$ is always trivial.
Category:16K20 

2. CJM 1999 (vol 51 pp. 69)
 Reichstein, Zinovy

On a Theorem of Hermite and Joubert
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 $\UD(n)$.
Categories:12E05, 16K20 
