Chapter

# Some Extensions and Applications of the Eisenstein Irreducibility Criterion

DOI: 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

\mathbbZ\mathbb{Z}
of integers proved by Eisenstein [9] in 1850 states as follows: Eisenstein Irreducibility Criterion. Let

F(x) = a0xn + a1xn-1 + ¼+ anF(x) = a_0x^n + a_1x^{n-1} + \ldots + a_n

be a polynomial with coefficient in the ring

\mathbbZ\mathbb{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,1 \leq i \leq n,
, and a

n

is not divisible by p
2, then F(x) is irreducible over the field

\mathbbQ\mathbb{Q}

of rational numbers.

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Available from: Sudessh Khanduja, Sep 05, 2014
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• ##### Article: Applications of the Newton Index to the Construction of Irreducible Polynomials
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ABSTRACT: We use properties of the Newton index associated to a polynomial with coefficients in a discrete valuation domain for generating classes of irreducible polynomials. We obtain factorization properties similar to the case of bivariate polynomials and we give new applications to the construction of families of irreducible polynomials over various discrete valuation domains. The examples are obtained using the package gp-pari.