| 摘要: | 令K 為一體, G 為一有限群。G 在K(xg | g ∈ G) 的作用為g · xh = xgh。K(G) = K(xg | g ∈ G)G。諾德問題考慮K(G)在K上是否有理(=純超越)。這個問題與反伽羅瓦問題有關。本計劃中我們將延續關於p-群的諾德問題的研究, 此處p為一質數。我們已知當|G| ≤ p4時,K(G) 在K上有理。2008年我們證明了當|G| = 32時,K(G)在K上有理。Bogomolov利用B0(G)建造出秩為|p6| 的群其B0(G) ̸= 0. 我們將秩為64的群分類, 並解決了其有理性問題, 除了G = (G; 64; i),241 ≤ i ≤ 245. Moravec 利用電腦計算證明B0(G) ̸= 0當G = G(243; i), 28 ≤ i ≤ 30. 本計劃中, 我們將研究其他秩為243群的有理性問題。我們並將回頭再看G = G(64; i),241 ≤ i ≤ 245群的有理性問題。最近, 康明昌及Hoshi 證明了存在秩為p5, p ≥ 3的群, 其B0(G) ̸= 0並得到一個這種群的集合與其個數。我們也將研究當|G| = p5且B0(G) ̸= 0 時, 這些群的有理性。
Let K be a eld, G be a nite group. Let G acts on K(xg | g ∈ G) by g · xh = xgh. The xed eld K(G) = K(xg | g ∈ G)G. Noether's problem asks whether K(G) is rational (=purely transcendental) over K. This problem is related to inverse Galois problem, the existence of generic G-Galois extensions. In this project, we shall continue our study on Noether's problem for p-groups, where p is a prime. It is known that K(G) is rational for |G| ≤ p4. We proved in 2008 that K(G) is rational for groups of order 25. For groups of order p6, Bogomolov constructed examples of groups of order p6 such that K(G) is not rational by using Bogomolov multiplier B0(G) as an obstruction,. We classi ed groups of order 26 and solved the rationality of K(G) except for groups G(64; i) for 241 ≤ i ≤ 245 where G(n; i) denote the i-th group of order n in GAP data base. Using computer, Moravec was able to prove that B0(G) ̸= 0 for G = G(243; i), 28 ≤ i ≤ 30. In this project, we shall study the rationality of G(243; i), i ̸= 28; 29; 30. We shall also investigate again rationality of K(G) where G = G(64; i), 241 ≤ i ≤ 245. Recently, Kang and Hoshi proved that for p ≥ 3, there exists groups of order p5 with B0(G) ̸= 0 and determined a class of such groups and the number in it. We shall also investigate rationality of K(G) for groups of order p5 with B0(G) ̸= 0. |
