Some Extensions and Applications of the Eisenstein Irreducibility Criterion

Summary

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 $\mathbb{Z}$ of integers proved by Eisenstein [9] in 1850 states as follows: Eisenstein Irreducibility Criterion. Let $F(x) = a_0x^n + a_1x^{n-1} + \ldots + a_n$ be a polynomial with coefficient in the ring $\mathbb{Z}$ 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 $1 \leq i \leq n,$, and a _n is not divisible by p ², then F(x) is irreducible over the field $\mathbb{Q}$ of rational numbers.

2010 Mathematics subject classification. 12E05, 12J10, 12J25.