{"paper":{"title":"Standard Polynomial Equations over Division Algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Adam Chapman, Casey Machen","submitted_at":"2015-08-01T23:23:16Z","abstract_excerpt":"Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\\phi(z)=z^n+c_{n-1} z^{n-1}+\\dots+c_0$ over $D$ a \"companion polynomial\" $\\Phi(z)$ of degree $n d$ with coefficients in $F$ whose roots are exactly the conjugacy classes of the roots of $\\phi(z)$. We explain how in case $D$ is a quaternion algebra, all the roots of $\\phi(z)$ can be recovered from the roots of $\\Phi(z)$. On the way, we also generalize certain theorems that were known for $\\mathbb{H}$ to any division algebra, such as the connection between the right eigenvalues of a matr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00173","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"}