Noether’s Problem and the Unramified Brauer Group for Groups of Order 64

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Title:	Noether’s Problem and the Unramified Brauer Group for Groups of Order 64
Authors:	Chu, Huah;Hu, Shou-Jen;Kang, Ming-Chang;Kunyavskii, Boris E.
Contributors:	淡江大學數學學系
Date:	2010
Issue Date:	2011-10-01 21:09:05 (UTC+8)
Publisher:	Oxford: Oxford University Press
Abstract:	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 p6 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.
Relation:	International Mathematics Research Notices 2010(12), pp.2329-2366
DOI:	10.1093/imrn/rnp217
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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