(untitled) (European Journal of Pure and Applied Mathematics, 2025)
The irreducibility of a polynomial over a finite field refers to whether the polynomial, with coefficients in that field, cannot be factored into nontrivial polynomials. It is surprising to discover that there exist very efficient but still little-known divisibility criteria. In this paper, we give some irreducibility criterions of a given polynomial with coefficients in F_q[X], were F_q is a finite field. The arguments can be extended to discuss our results, including potential applications or future research directions.