Definition:Irreducible Polynomial

Definition

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.

Examples

$x^2 - 2$ in Ring of Polynomials over Reals

Consider the polynomial:

$\map P x = x^2 - 2$

over the ring of polynomials $\R \sqbrk X$ over the real numbers.

Then $\map P x$ is not irreducible, as from Difference of Two Squares:

$x^2 - 2 \equiv \paren {x + \sqrt 2} \paren {x - \sqrt 2} $

$x^2 - 2$ in Ring of Polynomials over Rationals

Consider the polynomial:

$\map P x = x^2 - 2$

over the ring of polynomials $\Q \sqbrk X$ over the rational numbers.

Then $\map P x$ is irreducible.

$X^2 + 1$ in Ring of Polynomials over Reals

Let $\R \sqbrk X$ be the ring of polynomials in $X$ over the real numbers $\R$.

Then the polynomial $X^2 + 1$ is an irreducible element of $\R \sqbrk X$.

$x^2 + 1$ in Ring of Polynomials over Complex Numbers

Consider the polynomial:

$\map P x = x^2 + 1$

over the ring of polynomials $\C \sqbrk X$ over the complex numbers.

Then $\map P x$ is not irreducible, as:

$x^2 + 1 \equiv \paren {x + i} \paren {x - i}$

Also see

• Equivalence of Definitions of Irreducible Polynomial over Field
• Field is Integral Domain

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

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

• Results about irreducible polynomials can be found here.