Irreducible Polynomial/Examples/x^2 - 2 in Reals

Examples of Irreducible Polynomials

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:

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