Vol. 270, No. 1, 2014

 Recent Issues Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Vol. 299: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
Noether's problem for abelian extensions of cyclic $p$-groups

Ivo M. Michailov

Vol. 270 (2014), No. 1, 167–189
Abstract

Let $K$ be a field and $G$ a finite group. Let $G$ act on the rational function field $K\left(x\left(g\right):g\in G\right)$ by $K$-automorphisms defined by $g\cdot x\left(h\right)=x\left(gh\right)$ for any $g$, $h\in G$. Denote by $K\left(G\right)$ the fixed field $K{\left(x\left(g\right):g\in G\right)}^{G}$. Noether’s problem then asks whether $K\left(G\right)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K\left(G\right)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgroup of index $p$. The second main result is that $K\left(G\right)$ is rational over $K$ for any group of order ${p}^{5}$ or ${p}^{6}$ (where $p$ 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 $G$.)

 In loving memory of my dear mother
Keywords
Noether's problem, rationality problem, metabelian group actions
Mathematical Subject Classification 2010
Primary: 14E08, 14M20
Secondary: 13A50, 12F12
Milestones
Received: 18 January 2013
Accepted: 13 February 2013
Published: 2 August 2014
Authors
 Ivo M. Michailov Faculty of Mathematics and Informatics Shumen University “Episkop Konstantin Preslavsky” Universitetska Street 115 9700 Shumen Bulgaria