A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

Lhoussain El Fadil

doi:10.2478/tmmp-2023-0005

.blurhash-client-img { display: none !important; }

A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

Tatra Mountains Mathematical Publications

Volume 83 (2023): Issue 1 (February 2023)

By: Lhoussain El Fadil

Open Access

|Mar 2023

Abstract

The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be given, too.

References

[1] BROWN, R.: Roots of generalized Schönemann polynomials in Henselian extension fields, Indian J. Pure Appl. Math. 39 (2008), no. 5, 403–410.
Search in Google Scholar Back to article
[2] COHEN, D.—MOVAHHEDI, A.—SALINIER, A.: , Factorization over local fields and the irreducibility of generalized difference polynomials,Mathematika, 47 (2000), 173–196.10.1112/S0025579300015801
Search in Google Scholar Back to article
[3] DUMAS, G.: Sur quelques cas d’irréductibilité des polynômes à coeffcients rationnels, J. Math. Pures Appl. 6 (1906), 191–258.
Search in Google Scholar Back to article
[4] DEAJIM, A.—EL FADIL, L.—NAJIM, A.: On a theorem of Dedekind (submitted).
Search in Google Scholar Back to article
[5] EISENSTEIN, G.: Über die Irreduztibilit¨at und einige andere Eigenschaften der Gleichungen, von welcher die Theilung der ganzen Lemniscate abh¨angt, J. reine angew. Math. 39 (1850), 160–179.
Search in Google Scholar Back to article
[6]ELFADIL,L.: On Newton polygons techniques and factorization of polynomial over Henselian valued fields,J.of Algebra and its Appl. 19 (2020), DOI: S0219498820501881.10.1142/S0219498820501881
Search in Google Scholar Back to article
[7] GUARDIA, J.—MONTES, J.—NART, E.: Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364 (2012), no. 1, 361–416.
Search in Google Scholar Back to article
[8] HENSEL, K.: Untersuchung der Fundamentalgleichung einer Gattung für eine reelle Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante,J. Reine Angew. Math. 113 (1894), 61–83.
Search in Google Scholar Back to article
[9] JAKHAR, A.: On the factors of a polynomial, Bull. London Math. Soc. 52 (2020) 158–160.10.1112/blms.12315
Search in Google Scholar Back to article
[10] MURTY, M. R.: Prime numbers and irreducible polynomials, Amer. Math. Monthly 109 (2002), 452–458.10.1080/00029890.2002.11919872
Search in Google Scholar Back to article
[11] KHANDUJA, S. K.—SAHA, J.: On a generalization of Eisenstein’s irreducibility criterion,Mathematika 44 (1997), 37–41.10.1112/S0025579300011931
Search in Google Scholar Back to article
[12] MACLANE, S.: A construction for absolute values in polynomial rings,Trans.Amer. Math. Soc. 40 (1936), 363–395.10.1090/S0002-9947-1936-1501879-8
Search in Google Scholar Back to article
[13] NEUKIRCH, J.: Algebraic Number Theory. Springer-Verlag, Berlin, 1999.10.1007/978-3-662-03983-0
Search in Google Scholar Back to article
[14] ORE, O.: Newtonsche Polygone in der Theorie der algebraischen Körper, Math. Ann. 99 (1928), 84–117.10.1007/BF01459087
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2023-0005 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

Journal RSS Feed

Language: English

Page range: 51 - 60

Submitted on: Nov 12, 2022

Published on: Mar 7, 2023

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

irreducibility criterion,

Related subjects:

© 2023 Lhoussain El Fadil, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 83 (2023): Issue 1 (February 2023)