Equivalence of Definitions of Irreducible Polynomial over Field

Theorem

Let $K$ be a field.

The following definitions of the concept of Irreducible Polynomial are equivalent:

Definition 1

Let $R$ be an integral domain.

An irreducible polynomial over $R$ is an irreducible element of the polynomial ring $R \left[{X}\right]$.

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.

Proof

Note that by Field is Integral Domain, $K$ is indeed an integral domain.