Definition:Irreducible Polynomial

Definition 1
Let $R$ be an integral domain.

An irreducible polynomial over $R$ is an irreducible element of the polynomial ring $R[X]$.

Definition 2: for fields
Let $K$ be a field.

An irreducible polynomial over $K$ is a nonconstant polynomial over $K$ that is not the product of two polynomials of smaller degree.

Definition 3: for fields
Let $K$ be a field.

An irreducible polynomial over $K$ is a polynomial over $K$ that is not the product of two nonconstant polynomials.

Also see

 * Equivalence of Definitions of Irreducible Polynomial over Field


 * By Units of Ring of Polynomial Forms over Field, a polynomial in a single indeterminate with coefficients in a field is irreducible it is not a product of two polynomial forms of smaller degree.

This is not necessarily true for polynomials over a commutative ring.