Abel’s and Galois’s Proofs on Quintic Equations having no Radical Solutions are Invalid
In this work, the arguments of Abel and Galois that the quintic equations have no radical solutions are proven to be false. According to the work of Abel and Galois, it was commonly acknowledged around two hundred years ago that generic quintic equations had no radical solutions. For some quintic equations with peculiar forms, Tang Jianer and colleagues have shown that radical solutions exist. The theories of Abel and Galois are unable to explain these observations. On the other hand, Gauss and his colleagues proved the fundamental theorem of algebra. According to the theorem, there were n solutions for the n order equations, including radical and non-radical solutions. Abel and Galois’ opinions are contradicted by the fundamental theorem of algebra. Abel and Galois’ proofs should be re-examined for the reasons given above, and re-evaluated Abel’s original manuscript was painstakingly inspected by the author, who uncovered numerous serious inaccuracies. To illustrate that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel utilised the known solution of the cubic equation as a premise to compute the parameters of his equation. An expansion containing 14 things was written as seven, with the remaining seven items missing. Galois determined that the quintic equations had no radical solutions because the permutation group had no real normal subgroup, although these two issues had no necessary logic relationship. To demonstrate Galois’ theory, many algebraic connections between the roots of equations were used to replace the root itself. For the cubic and quartic equations, the efficiency of the radical extension group of automorphism mapping. This goes against the original idea of an automorphism mapping group, resulting in ambiguity and arbitrariness in the concept. The conclusion is that only symmetry exists for n order algebraic equations. There is no symmetry in Galois’ solvable group. Mathematicians should abandon Abel and Galois’ ideas and concentrate on finding radical solutions to high-order problems.
Department of Theoretical Physics and Pure Mathematics, Institute of Innovative Physics in Fuzhou, Fuzhou, China.
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