Year: 2025
Type: article
Abstract: This paper investigates the irreducibility properties of small-degree polynomials with specific pre-assigned coefficients over the field of rational numbers. Focusing on polynomials of degrees up to four, we analyze families of polynomials where some coefficients are fixed as , etc. We establish irreducibility of polynomials using combinatorial techniques and derive irreducibility conditions for others as: is irreducible over if and only if , is irreducible over except for and , is irreducible over if , is irreducible over when is any prime with , , and are irreducible over . In addition to the existing tools, the above mentioned results establish the irreducibility of small degree polynomial at first sight. Moreover, through our results, we extend the applicability of classical irreducibility criteria and provide new classes of irreducible polynomials. The combinatorial and analytical techniques employed offer a foundation for further research into higher-degree polynomials and their irreducibility over other fields.
Source: Mathematics and Statistics
Authors: Pradeep Maan, Deepak Deepak, Sangeeta Malik, Amit Sehgal
Institutions: Baba Mastnath University, Government of Haryana
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Funder: University Grants Commission
OpenAlex ID: W4410214433