# Irreducible Polynomial/Examples

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## Examples of Irreducible Polynomials

### $x^2 - 2$ 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:

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

### $8 x^3 - 6 x - 1$ in Ring of Polynomials over Rationals

Consider the polynomial:

$\map P x = 8 x^3 - 6 x - 1$

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

Then $\map P x$ is irreducible.