Unramified Brauer groups for groups of order p^5

Abstract

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g : g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\bC(G)$ is rational over $\bC$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\bC(G)$ over $\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer computing. We will give a theoretic proof of the following theorem (i.e. by the traditional bare-hand proof without using computers). Theorem. Let $p$ be any odd prime number. Then there is a group $G$ of order $p^5$ satisfying $B_0(G)\neq 0$ and $G/[G,G] \simeq C_p \times C_p$. In particular, $\bC(G)$ is not rational over $\bC$.