Noether's problem for cyclic groups of prime order

Ming-chang Kang Department of Mathematics, National Taiwan University

Number Theory mathscidoc:1702.24001

Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$ where $p$ is a prime number. Suppose that $G=\langle\sigma\rangle \simeq C_p$ acts on $k(x_0,\ldots,x_{p-1})$ by $k$-automorphisms defined as $\sigma:x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0$. Denote by $P$ the set of all prime numbers and define $P_0=\{p\in P:\bm{Q}(\zeta_{p-1})$ is of class number one$\}$ where $\zeta_n$ a primitive $n$-th root of unity in $\bm{C}$ for a positive integer $n$; $P_0$ is a finite set by \cite{MM}. Theorem. Let $k$ be an algebraic number field and $P_k=\{p\in P: p$ is ramified in $k\}$. Then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ for all $p\in P\backslash (P_0\cup P_k)$.
Noether's problem, rationality problem
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@inproceedings{ming-changnoether's,
  title={Noether's problem for cyclic groups of prime order},
  author={Ming-chang Kang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170217151106343878449},
}
Ming-chang Kang. Noether's problem for cyclic groups of prime order. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170217151106343878449.
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