Irreducibility of Polynomials up to Fourth Degree with Some Prescribed Coefficients

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 <img src=image/13440866_14.gif>, <img src=image/13440866_15.gif> etc. We establish irreducibility of polynomials using combinatorial techniques and derive irreducibility conditions for others as: <img src=image/13440866_01.gif> is irreducible over <img src=image/13440866_02.gif> if and only if <img src=image/13440866_03.gif>, <img src=image/13440866_04.gif> is irreducible over <img src=image/13440866_02.gif> except for <img src=image/13440866_05.gif> and <img src=image/13440866_06.gif>, <img src=image/13440866_07.gif> is irreducible over <img src=image/13440866_02.gif> if <img src=image/13440866_03.gif>, <img src=image/13440866_08.gif> is irreducible over <img src=image/13440866_16.gif> when <img src=image/13440866_09.gif> is any prime with <img src=image/13440866_10.gif>, <img src=image/13440866_11.gif>, <img src=image/13440866_12.gif> and <img src=image/13440866_13.gif> are irreducible over <img src=image/13440866_02.gif>. 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.