Abel's Impossibility Theorem

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We now demonstrate that a formula for the roots of a Monic polynomial p ∈ Poly3 (C) cannot be expressed by a formula f : {a0,a1,a2} → C3, for f analytic and no nested radicals.

Suppose otherwise, and consider a general Monic polynomial p ∈ Poly3 (C) with roots z1,z2,z3. As in (1), no such formula can express the roots of p. We can further construct paths that cyclically permute the roots z1,z2,z3 while inducing closed loops for each coefficient of p and for the value of f . The paths of √n f will either follow a closed loop or rotate by some angle θ. Applying the commutator the values of √n f undergo a rotation ∆θ = 0, while inducing a non-trivial permutation of the roots z1,z2,z3, contradiction.

Conclusion: Any formula would require use of nested radicals (Cardano’s Formula).