Abel's Impossibility Theorem

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Suppose otherwise, and let A = {a1,... a4}. Then, there exist functions f1,...,fk : A → C5 that expresses the roots of a general Monic polynomial p ∈ Poly5 (C). By previous case there must be at least N ≥ 3 levels of nested radicals.

Thus, suppose we have an expression of N ∈ N levels of nested roots. Since S5 is not soluble, then we know that there exists continuous paths such that their commutator induce a non-trivial permutation of the roots, while both the coefficients of p and fi, for 1 ≤ i ≤ k, follow a closed loop, contradiction.

Conclusion: One cannot construct an expression for the roots of a general p ∈ Poly5 (C) in terms of it’s coefficients, analytic functions, and radicals.