Journals Information
Mathematics and Statistics Vol. 13(2), pp. 120 - 125
DOI: 10.13189/ms.2025.130207
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Irreducibility of Polynomials up to Fourth Degree with Some Prescribed Coefficients
Pradeep Maan 1, Deepak 2,*, Sangeeta Malik 2, Amit Sehgal 3
1 Department of Mathematics, Faculty of Physical Sciences, Govt. College for Women Lakhan Majra, Rohtak-124514, Haryana, India
2 Department of Physical Sciences (Mathematics), Faculty of Sciences, Baba Mastnath University, Asthal Bohar-124021, Rohtak, Haryana, India
3 Department of Mathematics, Faculty of Physical Sciences, Pt. N.R.S Govt. College, Rohtak-124001, Haryana, India
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.
KEYWORDS
Irreducibility, Euler's Totient Function, Number of Divisors Function
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Pradeep Maan , Deepak , Sangeeta Malik , Amit Sehgal , "Irreducibility of Polynomials up to Fourth Degree with Some Prescribed Coefficients," Mathematics and Statistics, Vol. 13, No. 2, pp. 120 - 125, 2025. DOI: 10.13189/ms.2025.130207.
(b). APA Format:
Pradeep Maan , Deepak , Sangeeta Malik , Amit Sehgal (2025). Irreducibility of Polynomials up to Fourth Degree with Some Prescribed Coefficients. Mathematics and Statistics, 13(2), 120 - 125. DOI: 10.13189/ms.2025.130207.