On Noether's problem for cyclic groups of prime order

Let k be a field and G be a finite group acting on the rational function field k(x(g) vertical bar g is an element of G) by k-automorphisms h(x(g)) = x(hg) for any g, h is an element of G. Noether's problem asks whether the invariant field k(G) = k(x(g) vertical bar g is an element of 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 Q(C-p) is rational over Q. Only known 17 primes p for which Q(C-p) is rational over Q are p <= 43 and p = 61,67,71. We show that for primes p < 20000, Q(C-p) 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(C-p) is not (stably) rational over Q for undetermined 28 primes p out of 46.