{"paper":{"title":"Noether's Problem on Semidirect Product Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Huah Chu, Shang Huang","submitted_at":"2017-04-24T06:37:24Z","abstract_excerpt":"Let $K$ be a field, $G$ a finite group. Let $G$ act on the function field $L = K(x_{\\sigma} : \\sigma \\in G)$ by $\\tau \\cdot x_{\\sigma} = x_{\\tau\\sigma}$ for any $\\sigma, \\tau \\in G$. Denote the fixed field of the action by $K(G) = L^{G} = \\left\\{ \\frac{f}{g} \\in L : \\sigma(\\frac{f}{g}) = \\frac{f}{g}, \\forall \\sigma \\in G \\right\\}$. Noether's problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m \\rtimes C_n$ is a semidirect product of cyclic groups $C_m$ and $C_n$ with $\\mathbb{Z}[\\zeta_n]$ a unique factorization domain, and $K$ contains an $e$th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07053","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"}