Some Irreducible Polynomials Over a Finite Field

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 $\mathbb{F}_{q}[X]$, were $\mathbb{F}_{q}$ is a finite field. The arguments can be extended to discuss our results, including potential applications or future research directions.