Open Access
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(xggG) by k-automorphisms h(xg)=xhg for any g,hG. Noether’s problem asks whether the invariant field k(G)=k(xggG)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 Cp of prime order p, it is unknown whether there exist infinitely many primes p such that Q(Cp) is rational over Q. Only known 17 primes p for which Q(Cp) is rational over Q are p43 and p=61,67,71. We show that for primes p<20000, Q(Cp) is not (stably) rational over Q except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that Q(Cp) is not (stably) rational over 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

Vol.91 • No. 3 • March 2015
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