how can I characterize the minimal polynomials of all elements of this extension 'to show that they have all the distinct roots?
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 $f'\ne0$.)