Noether's problem for abelian extensions of cyclic \(p\)-groups (Q461306)

Let \(K\) be a field and let \(G\) be a finite group. Let \(L = K(x_g : g \in G)\) be the rational function field whose indeterminates are indexed by the elements of \(G\), and let \(G\) act on \(L\) as a group of \(K\)-automorphisms by \(g(x_h) = x_{gh}\) for any \(g, h \in G\). Noether's problem asks whether the subfield \(L^G\) of \(L\) fixed by \(G\) is rational over \(K\), i.e., a purely transcendental extension of \(K\). The paper under review starts with a survey of the existing literature on Noether's problem, and then establishes the rationality of \(L^G\) in new situations. In the two main results of the paper, \(G\) is assumed to be a \(p\)-group of order \(p^n\) and exponent \(p^e\) and having an abelian normal subgroup \(H\) such that \(G/H\) is cyclic of order \(p^t\), and \(K\) is assumed to be either of characteristic \(p\) or contains a primitive \(p^e\)-th root of unity. The first main result establishes the rationality of \(L^G\) when \(t=1\) under additional conditions, and the second main result establishes the rationality of \(L^G\) when \(n \leq 6\). In view of a known result by \textit{D. J. Saltman} [J. Algebra 258, No. 2, 507--534 (2002; Zbl 1099.13013)], the latter result does not hold if \(n=9\), leaving the cases \(n=7\) and \(n=8\) open.