Root(s) of a $3^\text{rd}$ degree polynomial over $
\Bbb Q$ are expressible
using radicals with the imaginary $i$. If a root $r$ is real,
by taking only the real part, $r$ is expressible using radicals over the
rational numbers.

Why not? See [*Casus irreducibilis*][1] on Wikipedia.


  [1]: https://en.wikipedia.org/wiki/Casus_irreducibilis