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:

Definition: Irreducible Polynomial

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.

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  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013