Anuj , Bishnoi ; Khanduja, Sudesh K.
(2010)
*Some extensions and applications of the Eisenstein irreducibility criterion*
Developments in Mathematics, 18
.
pp. 189-197.
ISSN 1389-2177

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Official URL: http://www.springerlink.com/content/gn134r74042t21...

Related URL: http://dx.doi.org/10.1007/978-1-4419-6211-9_10

## Abstract

Some generalizations of the classical Eisenstein and Schönemann Irreducibility Criteria and their applications are described. In particular some extensions of the Ehrenfeucht–Tverberg irreducibility theorem which states that a difference polynomial f(x) – g(y) in two variables is irreducible over a field K provided the degrees of f and g are coprime, are also given. The discussion of irreducibility of polynomials has a long history. The most famous irreducibility criterion for polynomials with coefficients in the ring Z of integers proved by Eisenstein [9] in 1850 states as follows: Eisenstein Irreducibility Criterion. Let F(x) = a_{0}x^{n} + a_{1}x^{n-1 }+ ^{...}+ a_{n} be a polynomial with coefficient in the ring Z of integers. Suppose that there exists a prime number p such that a 0 is not divisible by p, a i is divisible by p for 1 ≤i ≤n, and a n is not divisible by p ^{2} , then F(x) is irreducible over the field Q of rational numbers.

Item Type: | Article |
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ID Code: | 69944 |

Deposited On: | 19 Nov 2011 10:27 |

Last Modified: | 19 Nov 2011 10:27 |

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