{"paper":{"title":"Symbol $p$-Algebras of Prime Degree and their $p$-Central Subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Adam Chapman, Michael Chapman","submitted_at":"2016-07-18T19:15:27Z","abstract_excerpt":"We prove that the maximal dimension of a $p$-central subspace of the generic symbol $p$-algebra of prime degree $p$ is $p+1$. We do it by proving the following number theoretic fact: let $\\{s_1,\\dots,s_{p+1}\\}$ be $p+1$ distinct nonzero elements in the additive group $G=(\\mathbb{Z}/p \\mathbb{Z}) \\times (\\mathbb{Z}/p \\mathbb{Z})$; then every nonzero element $g \\in G$ can be expressed as $d_1 s_1+\\dots+d_{p+1} s_{p+1}$ for some non-negative integers $d_1,\\dots,d_{p+1}$ with $d_1+\\dots+d_{p+1} \\leq p-1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05243","kind":"arxiv","version":2},"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"}