# Definition:Irreducible Polynomial

## Definition

An **irreducible polynomial** is a polynomial which is not **reducible**.

### Definition 1

Let $R$ be an integral domain.

An **irreducible polynomial over $R$** is an irreducible element of the polynomial ring $R \sqbrk X$.

### 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.

## 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 Real Numbers

Consider the polynomial:

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

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

Then $\map P x$ is irreducible, as its factors:

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

are complex.

### $x^2 + 1$ 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

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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**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**irreducible polynomial** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**reducible polynomial** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**irreducible polynomial** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**reducible polynomial**