Let K be any field and G be a finite group acting on the rational function field K(xg: g ∈ G) by h · xg = xhg for any g, h ∈ G. Define K(G) = K(xg: g ∈ G)G. Noether's problem asks whether K(G) is rational (purely transcendental) over K. For any prime number p, Bogomolov shows that there is some group G of order p 6 with B0(G) ≠ 0, where B0(G) is the unramified Brauer group of ℂ(G), which is the subgroup of H2(G, ℚ/ℤ) consisting of cohomology classes whose restrictions to all bicyclic subgroups are zero. As a consequence, ℂ(G) is not rational over ℂ. In this paper, we will classify all the groups G of order 64 with B0(G) ≠ 0; for groups G satisfying B0(G) = 0, we will show that ℂ(G) is rational except possibly for five cases.
Bibliographical noteFunding Information:
Ming-chang Kang was partially supported by National Center for Theoretic Sciences (Taipei office). Parts of the work of this paper were done while he visited School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India.
Boris E. Kunyavskii was supported in part by the Minerva Foundation through the Emmy Noether Institute of Mathematics. A part of this work was done during his visit to the National Center of Theoretical Sciences (Taipei office) in September 2008. The support of these institutions is gratefully acknowledged.