{"paper":{"title":"Hilbert's fourteenth problem and field modifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Shigeru Kuroda","submitted_at":"2018-03-21T16:42:07Z","abstract_excerpt":"Let $k({\\bf x})=k(x_1,\\ldots ,x_n)$ be the rational function field, and $k\\subsetneqq L\\subsetneqq k({\\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\\cap k[x_1,\\ldots ,x_n]$ is finitely generated. Various counterexamples to this problem were already given, but the case $[k({\\bf x}):L]=2$ was open when $n=3$. In this paper, we study the problem in terms of the field-theoretic properties of $L$. We say that $L$ is minimal if the transcendence degree $r$ of $L$ over $k$ is equal to that of $A$. We show that, if $r\\ge 2$ and $L$ is minimal, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08002","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"}