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March 2015 On Noether’s problem for cyclic groups of prime order
Akinari Hoshi
Proc. Japan Acad. Ser. A Math. Sci. 91(3): 39-44 (March 2015). DOI: 10.3792/pjaa.91.39

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.

Citation

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Akinari Hoshi. "On Noether’s problem for cyclic groups of prime order." Proc. Japan Acad. Ser. A Math. Sci. 91 (3) 39 - 44, March 2015. https://doi.org/10.3792/pjaa.91.39

Information

Published: March 2015
First available in Project Euclid: 3 March 2015

zbMATH: 1334.12007
MathSciNet: MR3317750
Digital Object Identifier: 10.3792/pjaa.91.39

Subjects:
Primary: 11R18 , 11R29 , 12F12 , 13A50 , 14E08 , 14F22

Keywords: algebraic tori , Class number , cyclotomic field , Noether’s problem , rationality problem

Rights: Copyright © 2015 The Japan Academy

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Vol.91 • No. 3 • March 2015
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