{"paper":{"title":"On sums and products in a field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guang-Liang Zhou, Zhi-Wei Sun","submitted_at":"2018-07-02T15:36:05Z","abstract_excerpt":"In this paper we study sums and products in a field. Let $F$ be a field with ${\\rm ch}(F)\\not=2$, where ${\\rm ch}(F)$ is the characteristic of $F$. For any integer $k\\ge4$, we show that each $x\\in F$ can be written as $a_1+\\ldots+a_k$ with $a_1,\\ldots,a_k\\in F$ and $a_1\\ldots a_k=1$ if ${\\rm ch}(F)\\not=3$, and that for any $\\alpha\\in F\\setminus\\{0\\}$ we can write each $x\\in F$ as $a_1\\ldots a_k$ with $a_1,\\ldots,a_k\\in F$ and $a_1+\\ldots+a_k=\\alpha$. We also prove that for any $x\\in F$ and $k\\in\\{2,3,\\ldots\\}$ there are $a_1,\\ldots,a_{2k}\\in F$ such that $a_1+\\ldots+a_{2k}=x=a_1\\ldots a_{2k}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.01181","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"}