Equivalence of Definitions of Irreducible Polynomial over Field
Contents
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.
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