Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the
regular representation of $G$ over $k$. Then $G$ acts naturally on the rational
function field $k(V_{\rm reg})$ by $k$-automorphisms...Define $k(G)$ to be the
fixed field $k(V_{\rm reg})^G$. Noether's problem asks whether $k(G)$ is
rational (resp. stably rational) over $k$. When $k=\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\simeq C_8$ (the
cyclic group of order $8$), Jack Sonn proves that $\bQ(G)$ is not stably
rational over $\bQ$, which is a non-abelian extension of a theorem of
Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's
problem $\bQ(C_8)$. Using the method of cohomological invariants, we are able
to generalize Sonn's theorem as follows. Theorem. Let $G$ be a finite group and
$N$ $\lhd$ $G$ such that $G/N\simeq C_{2^n}$ with $n\geq 3$. If $k$ is a field
satisfying that ${\rm char}\,k=0$ and $k(\zeta_{2^n})/k$ is not a cyclic
extension where $\zeta_{2^n}$ is a primitive $2^n$-th root of unity, then
$k(G)$ is not stably rational (resp. not retract rational) over $k$. \end{abstract}(read more)