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 Unramified Brauer groups for groups of order p^5


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Let $k$ be any field, $G$ be a finite group acting on the rational function field $kx g : g\in G$ by $h\cdot x g=x {hg}$ for any $h,g\in G$. Define $kG=kx g : g\in G^G$. Noethers problem asks whether $kG$ is rational = purely transcendental over $k$. It is known that, if $\bCG$ is rational over $\bC$, then $B 0G=0$ where $B 0G$ is the unramified Brauer group of $\bCG$ over $\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B 0G=0$. 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 $B 0G eq 0$ and $G-G,G \simeq C p \times C p$. In particular, $\bCG$ is not rational over $\bC$.



Author: Akinari Hoshi; Ming-chang Kang

Source: https://archive.org/



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