Equivalence of Definitions of Irreducible Polynomial over Field

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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.