## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### On Noether’s problem for cyclic groups of prime order

Akinari Hoshi

#### Abstract

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_{g}\mid g\in G)$ by $k$-automorphisms $h(x_{g})=x_{hg}$ for any $g,h\in G$. Noether’s problem asks whether the invariant field $k(G)=k(x_{g}\mid g\in G)^{G}$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_{p}$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$. Only known 17 primes $p$ for which $\mathbf{Q}(C_{p})$ is rational over $\mathbf{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbf{Q}(C_{p})$ is not (stably) rational over $\mathbf{Q}$ for undetermined 28 primes $p$ out of 46.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 3 (2015), 39-44.

Dates
First available in Project Euclid: 3 March 2015

https://projecteuclid.org/euclid.pja/1425396669

Digital Object Identifier
doi:10.3792/pjaa.91.39

Mathematical Reviews number (MathSciNet)
MR3317750

Zentralblatt MATH identifier
1334.12007

#### Citation

Hoshi, Akinari. On Noether’s problem for cyclic groups of prime order. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 3, 39--44. doi:10.3792/pjaa.91.39. https://projecteuclid.org/euclid.pja/1425396669

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