An application of cohomological invariants

Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\mathrm{r}\mathrm{e}\mathrm{g}})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\mathrm{r}\mathrm{e}\mathrm{g}})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\mathrm{r}\mathrm{e}\mathrm{g}}{)}^G$. Noether’s problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\text{bQ}$ and $G$ contains a normal subgroup $N$ with $G\mathrm{/}H\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\text{bQ}(G)$ is not stably rational over $\text{bQ}$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether’s problem $\text{bQ}(C_8)$. Using the method of cohomological invariants, we are able to generalize Sonn’s theorem as follows. Theorem. Let $G$ be a finite group and $N$ $⊲$ $G$ such that $G\mathrm{/}N\simeq C_{2^n}$ with $n\ge 3$. If $k$ is a field satisfying that $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}k=0$ and $k({\zeta }_{2^n})\mathrm{/}k$ 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}