Is there any way to tell from the coefficients of a cubic equation (with integer coefficients) whether or not the solutions will all have a "simple" form. By "simple" I mean of the form
$a + b\sqrt{c}$
where $a, b, c \in \mathbb{Q}$
So, for example, the following would all be "simple" solutions:
$\bullet$ $2 + 3\sqrt{3}$
$\bullet$ $6$
$\bullet$ $\sqrt{5}$
$\bullet$ $\frac{1}{2} + \frac{2\sqrt{7}}{2}$
An example of a non-simple solution would be the solutions to $8x^3-5x^2+4x-4$ (try typing into wolfram alpha and see the resulting monstrosity)
If we assume that the cubic is irreducible and the coefficients are integers it cannot have a root of the form $a+b\sqrt{c}\ $ because the minimal polynomial of such a number over $\mathbb Q$ has at most degree $2$.