Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
irreducibility; Eisenstein criterion; polynomial
irreducibility; Eisenstein criterion; polynomial
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.
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. DOI 10.1215/kjm/1250520346 | MR 0981094 | Zbl 0689.12017
[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.
[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. DOI 10.1515/crll.1850.39.160 | MR 1578663
[4] Engler, A. J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics. Springer, New York (2005). DOI 10.1007/3-540-30035-X | MR 2183496 | Zbl 1128.12009
[5] Girstmair, K.: On an irreducibility criterion of M. Ram Murty. Am. Math. Mon. 112 (2005), 269-270. DOI 10.1080/00029890.2005.11920194 | MR 2125390 | Zbl 1077.11017
[6] Gouvêa, F. Q.: $p$-adic Numbers: An Introduction. Springer, New York (2003). DOI 10.1007/978-3-030-47295-5 | MR 4175370 | Zbl 1436.11001
[7] Jakhar, A.: On the factors of a polynomial. Bull. Lond. Math. Soc. 52 (2020), 158-160. DOI 10.1112/blms.12315 | MR 4072040 | Zbl 1455.11144
[8] Jakhar, A.: On the irreducible factors of a polynomial. Proc. Am. Math. Soc. 148 (2020), 1429-1437. DOI 10.1090/proc/14856 | MR 4069182 | Zbl 1446.12004
[9] Jakhar, A., Srinivas, K.: On the irreducible factors of a polynomial. II. J. Algebra 556 (2020), 649-655. DOI 10.1016/j.jalgebra.2020.02.045 | MR 4088446 | Zbl 1443.12002
[10] Jhorar, B., Khanduja, S. K.: Reformulation of Hensel's lemma and extension of a theorem of Ore. Manuscr. Math. 151 (2016), 223-241. DOI 10.1007/s00229-016-0829-z | MR 3532244 | Zbl 1351.12002
[11] Khanduja, S. K., Kumar, M.: Prolongations of valuations to finite extensions. Manuscr. Math. 131 (2010), 323-334. DOI 10.1007/s00229-009-0320-1 | MR 2592083 | Zbl 1216.12007
[12] Murty, M. Ram: Prime numbers and irreducible polynomials. Am. Math. Mon. 109 (2002), 452-458. DOI 10.1080/00029890.2002.11919872 | MR 1901498 | Zbl 1053.11020
[13] Schönemann, T.: Von denjenigen Moduln, welche Potenzen von Primzahlen sind. J. Reine Angew. Math. 32 (1846), 93-105 German. DOI 10.1515/crll.1846.32.93 | MR 1578516
[14] Weintraub, S. H.: A mild generalization of Eisenstein's criterion. Proc. Am. Math. Soc. 141 (2013), 1159-1160. DOI 10.1090/S0002-9939-2012-10880-9 | MR 3008863 | Zbl 1271.12001
[15] Weintraub, S. H.: A family of tests for irreducibility of polynomials. Proc. Am. Math. Soc. 144 (2016), 3331-3332. DOI 10.1090/proc/13033 | MR 3503701 | Zbl 1390.12001
Search
Partner of
