Chapter

Some Extensions and Applications of the Eisenstein Irreducibility Criterion

12/2009; 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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Anuj Bishnoi