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.

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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**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. - [Show abstract] [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. - [Show abstract] [Hide abstract]

**ABSTRACT:**We describe a method for constructing classes of bivariate polynomials which are irreducible over algebraically closed fields of characteristic zero. The constructions make use of some factorization conditions and apply to classes of polynomials that includes the generalized difference polynomials.