Given a polynomial $p(x)\in \mathbb{Q}[x]$, it is known that its roots can be obtained in terms of the coefficients of the polynomial by formulas involving the usual algebraic operations (addition, subtraction, multiplication, division), application of radicals (square roots, cube roots, etc), and application of automorphic functions.

What is known when we go up one level, to a polynomial $p(y)\in \mathbb{Q}(x)[y]$? Meaning, what operations on the coefficients (lying in $\mathbb{Q}(x)$) do we need in order to express the roots (lying in $\overline{\mathbb{Q}(x)}$)?