On a mild generalization of the Schönemann - Eisenstein - Dumas irreducibility criterion

ABSTRACT

We state a mild generalization of the classical Schönemann and Eisenstein- Dumas irreducibility criterion in ℤ[x] and provide an elementary proof. In the end of the paper, we also provide a concrete example of a polynomial which is irreducible by the main result of the paper but whose irreducibility does not follow from existing criteria.

KEYWORDS:

• Eisenstein criterion
• Eisenstein-Dumas irreducibility criterion
• polynomial irreduciblity
• Schönemann irreducibility criterion

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 12E05

Acknowledgment

The author is thankful to the anonymous referee for the valuable suggestions toward the improvement of the paper and the financial support from IISER Mohali is gratefully acknowledged by the author.

Notes

¹Let $g\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0$ be a polynomial with coefficients in ℤ. Suppose there exists a prime p whose exact power $p^{t_i}$ dividing a_i (where t_i = ∞ if a_i = 0), satisfy t_n = 0, $t_i∕(n-i)>(t_0∕n)$ for 0≤i≤n−1 and t₀,n are coprime. Then g(x) is irreducible over ℚ.