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.

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 $\UD(n)$.