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1. CJM 2000 (vol 52 pp. 1018)
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$-equivariant centers, into a $G$-variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 |
2. CJM 1999 (vol 51 pp. 69)
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 $\lambda^n + c_1 \lambda^{n-1} +
\cdots + c_{n-1} \lambda + c_n$ 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 |