It is not generally possible to determine the roots of a polynomial whose grade is bigger than 4 in terms of roots and basic operations. But I heard that it is possible to give a criteria whether a polynomial has such a solution.
For instance Wikipedia tells that the roots of a polynomial of degree $5$ are expressible in terms of roots and basic operations, if it is representable in the form
$x^5 + \frac{5\mu^4(4\nu + 3)}{\nu^2 + 1}x + \frac{4\mu^5(2\nu + 1)(4\nu + 3)}{\nu^2 + 1} = 0,$
where $\mu$ and $\nu$ are rational numbers.
Given the case that the polynomials roots are expressible in such a form, is it possible to give an algorithm that computes the solution in this form?
I am not an expert on that topic, just a student who is interested in maths.