Abel's Impossibility Theorem

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We now show that there is no formula for the roots z1,z2 of a general Monic polynomial p ∈ Poly2 (C) in terms of analytic (single valued) functions f ,g : {a0,a1} → C such that f(a0,a1) = z1 and g(a0,a1) = z2 for a general quadratic equation.

Suppose otherwise. Then, using Vieta’s formula we find that the coefficients a0,a1 given by a0 = z1z2 and a1 = −(z1+z2), which are symmetric expressions in z1,z2. Starting from distinct points z1,z2 we can continuously move them until they change places z1 → z2, z2 → z1. Under this motion: I Each of the coefficients a0,a1 follows a closed path, I The functions f(a0,a1) and g(a0,a1) follow closed paths. Contradicting the assumption that f and g follow the roots z1,z2, that interchanged places.

Conclusion: Any formula would require use of multi-valued function (Quadratic Formula).