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The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value

Published online by Cambridge University Press:  20 November 2018

Anca Iuliana Bonciocat  and
Nicolae Ciprian Bonciocat
Anca Iuliana Bonciocat
Affiliation:
Institute of Mathematics, of the Romanian Academy, Bucharest 014700, Romania e-mail: Anca.Bonciocat@imar.roNicolae.Bonciocat@imar.ro
Nicolae Ciprian Bonciocat
Affiliation:
Institute of Mathematics, of the Romanian Academy, Bucharest 014700, Romania e-mail: Anca.Bonciocat@imar.roNicolae.Bonciocat@imar.ro
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Abstract

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We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

Keywords

11C0811R09Estimates for polynomial rootsirreducible polynomials
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 511 - 520
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Ballieu, R., Sur les limitations des racines d’une équation algébrique. Acad. Roy. Belg. Bull. Cl. Sci. (5) 33(1947), 747–750.Google Scholar
[2] Bonciocat, N. C. and Zaharescu, A., Irreducible multivariate polynomials obtained from polynomials in fewer variables. J. Pure Appl. Algebra 212(2008), no. 10, 2338–2343.Google Scholar
[3] Brillhart, J., Filaseta, M. and Odlyzko, A., On an irreducibility theorem of A. Cohn, Canad. J. Math. 33 (1981) no. 5, 1055–1059.Google Scholar
[4] Cowling, V. F. and Thron, W. J., Zero-free regions of polynomials. Amer. Math. Monthly 61(1954), 682–687.Google Scholar
[5] Cowling, V. F. and Thron, W. J., Zero-free regions of polynomials. J. Indian Math. Soc. (N.S.) 20(1956), 307–310.Google Scholar
[6] Filaseta, M., A further generalization of an irreducibility theorem of A. Cohn. Canad. J. Math. 34(1982), no. 6, 1390–1395.Google Scholar
[7] Filaseta, M., Irreducibility criteria for polynomials with non-negative coefficients. Canad. J. Math. 40(1988), no. 2, 339–351.Google Scholar
[8] Fujiwara, M., Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung. Tôhoku Math. J. 10(1916), 167–171.Google Scholar
[9] Kojima, T., On a theorem of Hadamard's and its application. Tôhoku Math. J. 5(1914), 54–60.Google Scholar
[10] Kojima, T., The limits of the roots of an algebraic equation. Tôhoku Math. J. 11(1917), 119–127.Google Scholar
[11] Marden, M., Geometry of polynomials. Mathematical Surveys and Monographs No. 3, American Mathematical Society, Providence, RI, 1966.Google Scholar
[12] Perron, O., Algebra. II Theorie der algebraischen Gleichungen. Walter de Gruyter & Co., Berlin, 1951.Google Scholar
[13] Pólya, G., Szegö, G., Aufgaben und Lehrsätze aus der Analysis, Springer-Verlag, Berlin, 1964.Google Scholar
[14] Murty, M. Ram, Prime numbers and irreducible polynomials. Amer. Math. Monthly 109(2002), no. 5, 452–458.Google Scholar