{"paper":{"title":"The order of the reductions of an algebraic integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca","submitted_at":"2011-07-22T19:04:24Z","abstract_excerpt":"Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or, more generally, has some prescribed l-adic valuation). We evaluate the degree over K of extensions of the form K(\\zeta_m, \\sqrt[n]{a}) with n\\leq m, which are obtained by adjoining a root of unity of order l^m and the l^n-th roots of a, as this is needed for computing the above density."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4595","kind":"arxiv","version":4},"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"}