{"paper":{"title":"Rational invariants for subgroups of S_5 and S_7","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Baoshan Wang, Ming-chang Kang","submitted_at":"2013-08-02T07:29:44Z","abstract_excerpt":"Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\\ldots,x_n)$ via $k$-automorphisms defined by $\\sigma\\cdot x_i=x_{\\sigma(i)}$ for any $\\sigma\\in G$, any $1\\le i\\le n$. Theorem. If $n\\le 5$, then the fixed field $k(x_1,\\ldots,x_n)^G$ is purely transcendental over $k$. We will show that $\\bm{C}(x_1,\\ldots,x_7)^G$ is also purely transcendental over $\\bm{C}$ if $G$ is any transitive subgroups of $S_7$ other than $A_7$; a similar result is valid for solvable transitive subgroups of $S_{11}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0423","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"}