{"paper":{"title":"A note on extensions of $\\mathbb{Q}^{tr}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lukas Pottmeyer","submitted_at":"2014-08-27T13:41:17Z","abstract_excerpt":"In this note we investigate the behaviour of the absolute logarithmic Weil-height h on extensions of the field $\\mathbb{Q}^{tr}$ of totally real numbers. It is known that there is a gap between zero and the next smallest value of h on $\\mathbb{Q}^{tr}$, whereas in $\\mathbb{Q}^{tr}(i)$ there are elements of arbitrarily small positive height. We prove that all elements of small height in any finite extension of $\\mathbb{Q}^{tr}$ already lie in $\\mathbb{Q}^{tr}(i)$. This leads to a positive answer to a question of Amoroso, David and Zannier, if there exists a pseudo algebraically closed field wit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6411","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"}