Аннотация:
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 $\mathbb{C}(G)$ is rational over $\mathbb{C}$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\mathbb{C}(G)$ over $\mathbb{C}$ investigated by Saltman and Bogomolov. Bogomolov proves that for any prime number $p$, there is a $p$-group $G$ of order $p^6$ such that $B_0(G)$ is non-trivial and therefore $\mathbb{C}(G)$ is not rational over $\mathbb{C}$. He also shows that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. The latter result was disproved by Moravec for $p=3,5,7$ by the computer computing. The case for groups of order $32$ and $64$ was solved by Chu, Hu, Kang, Kunyavskii and Prokhorov. We will prove the following theorems. Theorem 1 (Hoshi, Kang and Kunyavskii). Let $p$ be any odd prime number and $G$ be a group of order $p^5$. If $G$ belongs to the isoclinism family $\Phi_{10}$ in R. James's classification of groups of order $p^5$ (“Math. Comp. 34 (1980) 613–637”), then $B_0(G)\neq 0$; in particular, $\mathbb{C}(G)$ is not rational over $\mathbb{C}$. On the other hand, if $G$ doesn't belong to the isoclinism family $\Phi_{6}$ or $\Phi_{10}$, then $B_0(G)= 0$. Theorem 2 (Chu, Hoshi, Hu and Kang). Let $G$ be a group of order $243$ with exponent $e$. Let $k$ be a field containing a primitive $e$-th root of unity. Then the followings are equivalent, (i) $k(G)$ is rational over $k$, (ii) $B_0(G)=0$, (iii) $G$ is not isomorphic to $G(243,i)$ with $28 \le i \le 30$ where $G(243,i)$ is the GAP code number for the $i$-th group of order $243$.
