Chapter

# Some Extensions and Applications of the Eisenstein Irreducibility Criterion

DOI: 10.1007/978-1-4419-6211-9_10 In book: Quadratic Forms, Linear Algebraic Groups, and Cohomology, pp.189-197

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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ABSTRACT: We study the irreducibility of general bivariate polynomials over algebraically closed fields of characteristic zero. We obtain factorization conditions in terms of the degree index and we deduce the irreducibility for classes of polynomials that include that of quasi-generalized difference polynomials.
No preview · Article · Jan 2013 · Bulletin mathématiques de la Société des sciences mathématiques de Roumanie
• ##### Article: On the irreducibility of bivariate polynomials
[Hide abstract]
ABSTRACT: We study the irreducibility of general bivariate polynomials over algebraically closed fields of characteristic zero. We obtain factorization conditions in terms of the degree index and we deduce the irreducibility for classes of polynomials that include that of quasi-generalized difference polynomials.
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