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On Noether’s problem for cyclic groups of prime order. (English) Zbl 1334.12007

Summary: 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 \(\mathbb{Q}(C_{p})\) is rational over \(\mathbb{Q}\). Only known 17 primes \(p\) for which \(\mathbb{Q}(C_{p})\) is rational over \(\mathbb{Q}\) are \(p\leq 43\) and \(p=61,67,71\). We show that for primes \(p< 20000\), \(\mathbb{Q}(C_{p})\) is not (stably) rational over \(\mathbb{Q}\) except for affirmative 17 primes and undetermined 46 primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that \(\mathbb{Q}(C_{p})\) is not (stably) rational over \(\mathbb{Q}\) for undetermined 28 primes \(p\) out of 46.

MSC:

12F20 Transcendental field extensions
14E08 Rationality questions in algebraic geometry
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants

Software:

PARI/GP
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References:

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