Abel’s and Galois’s Proofs on Quintic Equations having no Radical Solutions are Invalid

Abstract

The proofs of Abel and Galois that the quintic equations have no radical solutions are shown to be erroneous in this study. Around two hundred years ago, it was widely accepted that general quintic equations had no radical solutions, according to the work of Abel and Galois. Tang Jianer and colleagues recently demonstrated that radical solutions exist for some quintic equations with special forms. Abel's and Galois' ideas are unable to explain these findings. Gauss and his colleagues, on the other hand, proved the fundamental theorem of algebra. There were n solutions for the n order equations, including radical and non-radical solutions, according to the theorem. The fundamental theorem of algebra contradicts Abel and Galois' beliefs. The proofs of Abel and Galois should be re-examined and re-evaluated for the reasons stated above. The author meticulously examined Abel's original manuscript and discovered several severe errors. Abel used the known solution of the cubic equation as a premise to compute the parameters of his equation in order to prove that the general solution of algebraic equation he suggested was effective for the cubic equation. An expansion with 14 items was written as 7 items, the other 7 items were missing. Based on the permutation group  had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In Galois' theory, several algebraic relations among the roots of equations were employed to replace the root itself in order to illustrate the efficiency of radical extension group of automorphism mapping for the cubic and quartic equations. This went against the initial notion of an automorphism mapping group, resulting in conceptual ambiguity and arbitrariness. It is concluded that there is only the  symmetry for the n order algebraic equations. The symmetry of Galois’s solvable group does not exist. Mathematicians should get rid of the constraints of Abel and Galois’s theories, keep to looking for the radical solutions of high order equations.