Gras, Georges On the order modulo \(p\) of an algebraic number. (English. French summary) Zbl 1428.11183 J. Théor. Nombres Bordx. 30, No. 1, 307-329 (2018). Let \(K/\mathbb Q\) be a Galois extension of degree \(n\) with Galois group \(G\) and let \(\eta\in K\) generate a multiplicative \(\mathbb Z[G]\)-module of rank \(n\). For an unramified prime ideal \(\mathfrak p\) of \(K\) lying over the prime \(p\) let \(o_{\mathfrak p}(\eta)\) be the order of \(\eta\) mod \(\mathfrak p\), and let \(o_p(\eta)\) be the order of \(\eta\) mod \(p\). Let \(h\) be the order of a cyclic subgroup of \(G\). The author shows (Theorem 2.1) that for all sufficiently large primes \(p\) with residue degree \(h\) and prime ideals \({\mathfrak p}\mid p\) the minimal integer \(k\ge1\) such that \(\eta^k\) is congruent to a root of unity mod \(\mathfrak p\) does not divide \((p^h-1)/\Phi_d(p)\) for \(d\mid h\), where \(\Phi_d(X)\) is the \(d\)-th cyclotomic polynomial. This implies the corresponding assertion for the numbers \(o_p(\eta)\) and \(o_{\mathfrak p}(\eta)\). In Theorem 4.1 a lower bound for this number \(k\) is provided.The remainder of the paper is devoted to heuristical study of the probability of the inequality \(o_p(\eta) <p\). The author proposes the following conjecture: If \(\eta\) is as above, the for all not totally split primes \(p\) one has \(o_p(\eta)>p\), with finitely many exceptions. Some numerical evidence of the truth of this is also given. Reviewer: Władysław Narkiewicz (Wrocław) MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R16 Cubic and quartic extensions Keywords:algebraic numbers; order mod \(p\); Frobenius automorphisms; probabilistic number theory Software:PARI/GP × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Georges Gras, Class Field Theory: from theory to practice, Springer Monographs in Mathematics, Springer, 2005 · Zbl 1019.11032 · doi:10.1007/978-3-662-11323-3 [2] Georges Gras, Les \(\theta \)-régulateurs locaux d’un nombre algébrique : Conjectures \(p\)-adiques, Can. J. Math.68 (2016), p. 571-624 · Zbl 1351.11033 · doi:10.4153/CJM-2015-026-3 [3] Georges Gras, Étude probabiliste des quotients de Fermat, Funct. Approximatio, Comment. Math.54 (2016), p. 115-140 · Zbl 1407.11094 · doi:10.7169/facm/2016.54.1.9 [4] Pieter Moree, Artin’s Primitive Root Conjecture - A Survey, Integers12 (2012), p. 1305-1416 · Zbl 1271.11002 · doi:10.1515/integers-2012-0043 [5] Władysław Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer Monographs in Mathematics, Springer, 2004 · Zbl 1159.11039 [6] Gérald Tenenbaum, Introduction à la Théorie Analytique et Probabiliste des Nombres, Belin, 2015 · Zbl 0880.11001 [7] The PARI Group, “PARI/GP version 2.9.0” 2016, available from [8] Lawrence C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83, Springer, 1997 - ISSN : 2118-8572 (online) 1246-7405 (print) - Société Arithmétique de Bordeaux · Zbl 0966.11047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.