{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:49:48Z","timestamp":1776836988645,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"343","license":[{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a number field, and let\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a finitely generated subgroup of\n \n \n \n \n K<\/mml:mi>\n \n \u00d7\n \n <\/mml:mo>\n <\/mml:msup>\n K^\\times<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes\n \n \n \n \n p<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that the order of\n \n \n \n \n (<\/mml:mo>\n G<\/mml:mi>\n mod<\/mml:mo>\n \n p<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n (G\\bmod \\mathfrak p)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes\n \n \n \n \n p<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for which the order is\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -free, and those for which the order has a prescribed\n \n \n \n \n \u2113\n \n <\/mml:mi>\n \\ell<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -adic valuation for finitely many primes\n \n \n \n \n \u2113\n \n <\/mml:mi>\n \\ell<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . An additional condition on the Frobenius conjugacy class of\n \n \n \n \n p<\/mml:mi>\n <\/mml:mrow>\n \\mathfrak p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.\n <\/p>","DOI":"10.1090\/mcom\/3848","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T09:13:24Z","timestamp":1680081204000},"page":"2281-2305","source":"Crossref","is-referenced-by-count":0,"title":["Divisibility conditions on the order of the reductions of algebraic numbers"],"prefix":"10.1090","volume":"92","author":[{"given":"Pietro","family":"Sgobba","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,5,3]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1142\/S1793042123500124","article-title":"Divisibility of reduction in groups of rational numbers II","volume":"19","author":"Abdullah, H. O.","year":"2023","journal-title":"Int. J. 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