Following the definition of irreducible elements of a general ring $R,$ the following definition specifies what an irreducible polynomial in the ring $R[X]$ is:
Let $(R, + ,\cdot)$ be an integral domain. An irreducible polynomial $p\in R[X]$ is a polynomial of degree $\deg(p)\ge 1$ such that if $p=gh$ is factorization with polynomials $g,h\in R[X],$ then either $g$ or $h$ is a unit. In other words, either $g\in R^\ast$ or $h\in R^\ast,$ where $R^\ast$ is the group of units.