{"paper":{"title":"Symbol Length of $p$-Algebras of Prime Exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Adam Chapman","submitted_at":"2016-02-22T19:30:22Z","abstract_excerpt":"We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree $p$ over a field $F$ with $\\operatorname{char}(F)=p$ is a finite integer $d$ greater than 1 then the symbol length of $p$-algebras of exponent $p$ over $F$ is bounded from above by $\\left \\lceil \\frac{d-1}{p} \\right \\rceil-1$, and show that every two tensor products of symbol algebras of lengths $k$ and $\\ell$ with $(k+\\ell) p \\geq d-1$ can be modified so that they share a common slot. For $p=2$, we obtain an upper bound of $\\frac{u(F)}{2}-1$ for the symbol length, which is sharp when $I_q^3 F="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06901","kind":"arxiv","version":3},"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"}