Show that $\mathbb{Q}[\sqrt{2},\sqrt{3}] $ is separable

Question

how can I characterize the minimal polynomials of all elements of this extension 'to show that they have all the distinct roots?

Answer 1

More generally, we have this:

Every irreducible polynomial over a field of characteristic zero has distinct roots.

Indeed, $f'\ne0$ and $\deg f' < \deg f$ imply $\gcd(f,f')=1$.

(The part that can fail in positive characteristic is $f'\ne0$.)