In 1824, Abel correctly proved that the general quintic equation is root insoluble for the first time. A more detailed proof was published in the first issue of 1826 "crell" magazine, entitled "Proof of the impossibility of algebraic solutions of general equations higher than quartic". In this paper, Abel discusses and corrects the defects in Ruffini's argument. Rufini's "proof". Therefore, it can't work under the expansion of the basic definition domain and the definition domain determined by the coefficients of the known equation. In addition, Rufini's "proof" also used an unproven key proposition, which was later called Abel theorem. This theorem says that if an algebraic equation can be solved by roots, then every root in the expression of roots can be expressed as the root of the equation and some rational functions of unit roots. Abel used this theorem to prove that general equations higher than quartic cannot have root solutions.

The Abel theorem mentioned above is also the idea of "permutation group"

As upstairs said, Abel's theorem should be like this! Details are as follows:

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