{"paper":{"title":"Noether's problem for the groups with a cyclic subgroup of index 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AC","authors_text":"Ivo M. Michailov, Jian Zhou, Ming-chang Kang","submitted_at":"2011-08-17T01:50:48Z","abstract_excerpt":"Let $G$ be a finite group and $k$ be a field. 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 $g,h\\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\\in G)^G$ is rational (i.e. purely transcendental) over $k$. Theorem 1. If $G$ is a group of order $2^n$ ($n\\ge 4$) and of exponent $2^e$ such that (i) $e\\ge n-2$ and (ii) $\\zeta_{2^{e-1}} \\in k$, then $k(G)$ is $k$-rational. Theorem 2. Let $G$ be a group of order $4n$ where $n$ is any positive integer (it is unnecessary to assume that $n$ is a power of 2). Ass"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3379","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}