Consider the following quartic equation:
$$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$
By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational number, my question is simply if we assume $r$ is an algebraic number (which is a more general set of numbers that contains all rational numbers), can this equation still possess four radical roots
The roots of the equation are radical if and only if $r$ itself can be expressed by radicals.