Irreducible Polynomial/Examples

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 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:

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

$8 x^3 - 6 x - 1$ 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.

$x^3 + x^2 - 2 x - 1$ in Ring of Polynomials over Rationals

Irreducible Polynomial/Examples/x^3 + x^2 - 2 x - 1 in Rationals