Noether's problem for abelian extensions of cyclic $p$-groups

Abstract

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G{)}^G$. Noether's problem then asks whether $K(G)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K(G)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgoup of index $p$. The second main result is that $K(G)$ is rational over $K$ for any group of order $p^5$ or $p^6$ ($p$ is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if $charK\ne p$ then $K$ contains a primitive $p^e$-th root of unity, where $p^e$ is the exponent of $G$.)

arXiv admin note: text overlap with arXiv:1201.5555; and text overlap with arXiv:1201.5555, arXiv:0911.1162 by other authors

Cited by in corpus (3)

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