Noether's problems for groups of order 243

2014/03/03 by Chu, Huah, Hoshi, Akinari, Hu, Shou-Jen +1
#13A50 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1403.0318

Abstract

Let k be any field, G be a finite group. Let G act on the rational function field k(x_g:g∈ G) by k-automorphisms defined by h⋅ x_g=x_hg for any g,h∈ G. Denote by k(G)=k(x_g:g∈ G)^G the fixed field. Noether's problem asks, under what situations, the fixed field k(G) will be rational (= purely transcendental) over k. According to the data base of GAP there are 10 isoclinism families for groups of order 243. It is known that there are precisely 3 groups G of order 243 (they consist of the isoclinism family Φ₁₀) such that the unramified Brauer group of \bmC(G) over \bmC is non-trivial. Thus \bmC(G) is not rational over \bmC. We will prove that, if ζ₉ ∈ k, then k(G) is rational over k for groups of order 243 other than these 3 groups, except possibly for groups belonging to the isoclinism family Φ₇.

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