Is $x^2-2$ irreducible over R and Q? I'm not sure if it is valid to say that $x^2 - 2$ can be factorised to $2\cdot\left(\frac 12x^2 - 1\right)$ for it to be reducible in Q.
Though I know $(x + \sqrt{2})(x - \sqrt{2})$ works in the reals.

The polynomial is ofcourse reducible over $\mathbb{R}$ as it can be written as a product of polynomials of (strictly) smaller degrees.
Again, by Eisenstein's criteria it is irreducible over $\mathbb{Q}$
