令K 為一體, G 為一有限群。對於G的表現V , G 作用於有理函數體K(V )。有理性 問題就是要決定固定體K(V )G 是否有理(=純超越)。諾德問題為當V 為正則表現時的特例。 若我們以K(G) 表示其固定體。諾德問題就是要決定K(G) 在K 之上是否為有理的(=純超 越。) 這是代數幾何中有理性問題的特殊情形。幾何上就是單有理的代數簇何時為有理的問題。 如果體上有足夠的單位根, 我們已知對於秩為pn, n · 5的群G ,K(G) 是有理的。 Bogomolov 利用Bogomolov multiplier B0(G) 證明當秩為p6 時, 反例存在。他建議可以 在jGj = p6 時將B0(G) 6= 0的群分類。我們將從p = 2 時開始考慮這個問題。我們也將繼續 研究當群的秩為p5 而p為奇質數時諾德問題。 Let K be any field and G be a finite group. For a given representation V , G acts on the rational function field K(V ). Rationality problem is to determine whether the fixed field K(V )G is rational (=purely transcendental) over K. Noether’s problem is a special case of rationality problem when the representation is the regular representation. In this case we denote the fixed field by K(G). Geometrically it is a special case of the question: When will a unirational variety be a rational variety ? For fields containing enough roots of unity, it is known that K(G) is rational for groups of order pn for n · 5. Bogomolov used Bogomolov multiplier B0(G) to prove that counterexample exists for groups of order p6. He suggested to classify groups of order p6 with B0(G) 6= 0. We will look into this problem starting from p = 2. We will also further study the problem for groups of order p5 for odd prime p.
