Algorithm to find the exact roots of solvable high-order polynomials?

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.

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Yes, it’s called Galois Theory.

While one cannot express the terms of polynomials of degree 5 and higher with basic operations and roots (Abel-Ruffini), it is possible to explicitly calculate the roots of polynomials of degree five and higher if you use other tools.

See this MO question.

Yes, assuming an expression for a root exists, there is a simple algorithm that will find it: define an ordering on all such expressions (e.g. a lexical ordering), and test each one until you find a root of the input polynomial.