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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 $\overline{\mathbb{Q}(x)}$)?

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This is within the scope of a classical theorem on Riemann surfaces. See en.wikipedia.org/wiki/Uniformization_theorem. –  Charles Matthews Apr 10 '11 at 8:19
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+1 to Dror for editing. –  Gerry Myerson Apr 10 '11 at 12:30
    
Dror:Thank you for your editing.I think I had described my question badly –  XL _at_China Apr 10 '11 at 14:31
    
:) I edited because I didn't want to see this question closed. In fact, I still want to see the question answered! I don't know much about this theory, let alone how the Uniformization Theorem solves it. Since I'm a graduate student, I think the normal MO rules apply and say that someone should answer. So, please answer! –  Dror Speiser Apr 10 '11 at 19:54
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