TITLE:
A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions
AUTHORS:
Xiaochun Mei
KEYWORDS:
Quintic Equation, Gauss Basic Theorem of Algebra, Radical Solution, Abel’s Theory, Galois’s Theory, Solvable Group, Lagrange’s Resolvents
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.10 No.9,
September
14,
2020
ABSTRACT: It is proved in this paper that Abel’s and Galois’s proofs
that the quintic equations have no radical solutions are invalid. Due to Abel’s
and Galois’s work about two hundred years ago, it was generally accepted that
general quintic equations had no radical solutions. However, Tang Jianer et al. recently prove that there are radical solutions for some quintic
equations with special forms. The theories of Abel and Galois cannot explain
these results. On the other hand, Gauss et al. proved the fundamental theorem of algebra. The theorem declared that there
were n solutions for the n degree equations, including the
radical and non-radical solutions. The theories of Abel and Galois contradicted
with the fundamental theorem of algebra. Due
to the reasons above, the proofs of Abel and Galois should be re-examined and
re-evaluated. The author carefully analyzed the Abel’s
original paper and found some serious mistakes. In order to prove that the general
solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solutions
of cubic equation as a premise to
calculate the parameters of his equation. Therefore, Abel’s proof is a logical
circular argument and invalid. Besides, Abel confused the variables with the coefficients
(constants) of algebraic equations. An expansion with 14 terms was written as 7
terms, 7 terms were missing. We prefer to consider
Galois’s theory as a hypothesis rather than a proof. Based on that permutation
group S5 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 order to prove the effectiveness of radical extension group of automorphism mapping
for the cubic and quartic equations, in the
Galois’s theory, some algebraic relations among the roots of equations
were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and
arbitrariness. For the general cubic and quartic algebraic equations, the
actual solving processes do not satisfy the tower structure of Galois’s
solvable group. The resolvents of cubic and quartic equations are proved to
have no symmetries of Galois’s soluble
group actually. It is invalid to use the solvable group theory to judge whether
the high degree equation has a radical solution. The conclusion of this paper
is that there is only the Sn symmetry for the n degree 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 looking for the
radical solutions of high degree equations.