Degree three unramified cohomology groups

Abstract

Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G) by k-automorphisms defined by h⋅xg=xhg for any g,h∈G. Denote by k(G)=k(xg:g∈G)G, the fixed subfield. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. The unramified Brauer group Brnr(C(G)) and the unramified cohomology Hnr3(C(G),Q/Z) are obstructions to the rationality of C(G) (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G such that |G|=p12, Brnr(C(G))={0}, but Hnr3(C(G),Q/Z)≠{0}; thus C(G) is not stably C-rational [12]. Using Peyre's method, we are able to find groups G with |G|=p9 where p is an odd prime number such that Brnr(C(G))={0}, Hnr3(C(G),Q/Z)≠{0}.