Jakhar, Anuj On the irreducible factors of a polynomial over a valued field. (English) Zbl 07893386 Czech. Math. J. 74, No. 2, 367-375 (2024). Summary: We explicitly provide numbers \(d\), \(e\) such that each irreducible factor of a polynomial \(f(x)\) with integer coefficients has a degree greater than or equal to \(d\) and \(f(x)\) can have at most \(e\) irreducible factors over the field of rational numbers. Moreover, we prove our result in a more general setup for polynomials with coefficients from the valuation ring of an arbitrary valued field. MSC: 12E05 Polynomials in general fields (irreducibility, etc.) 11R09 Polynomials (irreducibility, etc.) 12J10 Valued fields Keywords:irreducibility; Eisenstein criterion; polynomial × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Alexandru, V.; Popescu, N.; Zaharescu, A., A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ. 28 (1988), 579-592 · Zbl 0689.12017 · doi:10.1215/kjm/1250520346 [2] Dumas, G., Sur quelques cas d’irréductibilité des polynomes á coefficients rationnels, J. Math. Pures Appl. 6 (1906), 191-258 French \99999JFM99999 37.0096.01 · JFM 37.0096.01 [3] Eisenstein, G., Über die Irreductibilität und einige andere Eigenschaften der Gleichungen, von welcher die Theilung der ganzen Lemniscate abhängt, J. Reine Angew. Math. 39 (1850), 160-179 German · ERAM 039.1073cj · doi:10.1515/crll.1850.39.160 [4] Engler, A. J.; Prestel, A., Valued Fields, Springer Monographs in Mathematics. Springer, New York (2005) · Zbl 1128.12009 · doi:10.1007/3-540-30035-X [5] Girstmair, K., On an irreducibility criterion of M. Ram Murty, Am. Math. Mon. 112 (2005), 269-270 · Zbl 1077.11017 · doi:10.1080/00029890.2005.11920194 [6] Gouvêa, F. Q., \(p\)-adic Numbers: An Introduction, Springer, New York (2003) · Zbl 1436.11001 · doi:10.1007/978-3-030-47295-5 [7] Jakhar, A., On the factors of a polynomial, Bull. Lond. Math. Soc. 52 (2020), 158-160 · Zbl 1455.11144 · doi:10.1112/blms.12315 [8] Jakhar, A., On the irreducible factors of a polynomial, Proc. Am. Math. Soc. 148 (2020), 1429-1437 · Zbl 1446.12004 · doi:10.1090/proc/14856 [9] Jakhar, A.; Srinivas, K., On the irreducible factors of a polynomial. II, J. Algebra 556 (2020), 649-655 · Zbl 1443.12002 · doi:10.1016/j.jalgebra.2020.02.045 [10] Jhorar, B.; Khanduja, S. K., Reformulation of Hensel’s lemma and extension of a theorem of Ore, Manuscr. Math. 151 (2016), 223-241 · Zbl 1351.12002 · doi:10.1007/s00229-016-0829-z [11] Khanduja, S. K.; Kumar, M., Prolongations of valuations to finite extensions, Manuscr. Math. 131 (2010), 323-334 · Zbl 1216.12007 · doi:10.1007/s00229-009-0320-1 [12] Murty, M. Ram, Prime numbers and irreducible polynomials, Am. Math. Mon. 109 (2002), 452-458 · Zbl 1053.11020 · doi:10.1080/00029890.2002.11919872 [13] Schönemann, T., Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. Reine Angew. Math. 32 (1846), 93-105 German · ERAM 032.0910cj · doi:10.1515/crll.1846.32.93 [14] Weintraub, S. H., A mild generalization of Eisenstein’s criterion, Proc. Am. Math. Soc. 141 (2013), 1159-1160 · Zbl 1271.12001 · doi:10.1090/S0002-9939-2012-10880-9 [15] Weintraub, S. H., A family of tests for irreducibility of polynomials, Proc. Am. Math. Soc. 144 (2016), 3331-3332 · Zbl 1390.12001 · doi:10.1090/proc/13033 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.