On two conjectures of irreducible polynomials

Abstract

In this paper, we give a new proof using Newton polygons for an irreducibility criterion established by Singh and Kumar. Then we prove a conjecture of Koley and Reddy. In addition, we give a new proof for the statement that $x^n\pm c{x}^{n-1}\pm c$ with $c⩾2$ is irreducible unless $(n,c)=(2,4)$, which is a conjecture of Harrington.

Keywords:

• Irreducible polynomials
• Newton polygon
• Rouché’s Theorem

2020 Mathematics Subject Classification:

• 11R09
• 11C08

Acknowledgments

We would like to thank the referees for detailed comments. Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 12171163).