Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$ individually.

Question

Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$, individually.

I have a bit of a start, but need some help.

Answer 1

Hint: Factor $$f(x) = (x^2 - 3)(x^2 - 2)$$

• Show that over $\Bbb{Q}$, these are irreducible; begin by noting that a second degree polynomial (over any field) is irreducible $\iff$ it has no zeros in the field. Next realize something about $\sqrt{2}$ and $\sqrt{3}$.

• Factor the polynomial over $\mathbb{R}$.