In 1824, Abel proved that there is no general formula for solving algebraic equations with five or more times. This proof is written in "in algebra, the so-called equation has a radical solution (algebraic solvability), that is, the solution of this equation can be expressed by finite operations such as addition, subtraction, multiplication and division by integer powers." The solution of algebraic equations has been algebra since1the first half of 6th century. The general solutions of cubic and quartic equations were solved by several mathematicians in Italy. In the next few hundred years, algebra mainly devoted itself to solving equations of quintic or even higher order, but it never succeeded. For the theory of equations, Lagrange systematically studied the properties of the roots of equations (1770), and correctly pointed out that the permutation and replacement theory of the roots of equations is the key to solving algebraic equations. Thereby realizing the transformation of algebraic thinking mode. Although Lagrange did not completely solve the problem of solving higher-order equations, his thinking method inspired future generations. 1799, P. Ruffini first proved the insolubility of general equations higher than quartic, but its "proof" was flawed. Two years later, Gauss solved the theoretical problem of the solvability of the secant equation. The work of Lagrange and Gauss is Abel's research. In his freshman year, he began to study Gauss's thesis Arithmetic. Later, he learned about Cauchy's achievements in permutation theory. However, he didn't know Rufini's work at that time. It is against this background that Abel thinks about the theoretical problem of solvability of algebraic equations.

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 Rufini'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"

When he was further thinking about which equations (such as x n-1= 0) can be solved by roots, Abel proved the following theorem: for an equation of any degree, if all the roots of the equation can be represented rationally by one of them (we use x to represent it), And if any two roots Q(x) and Q 1(x) (X) satisfy the relation QQ 1(x)=Q 1Q(x), then the equation under consideration is always algebraic, in other words, the root xi=Q 1(Xi).

Abel's legacy contains an unfinished manuscript, Surla Ré solution Algé brique des Fonts (1839). This paper describes the development of equation theory and discusses the solvability of special equations again. It paved the way for the publication of Galois's posthumous works. In the preface, Abel hinted at an important way of thinking. He believes that the existence of the solution should be proved before solving the equation, so that the whole process can avoid "computational complexity". In the theoretical study of solvability of algebraic equations, he also put forward a research program, that is, his work needs to solve two kinds of problems: one is to construct algebra of arbitrary numbers. The second is to determine whether the known equation can be solved by roots. He tried to describe the characteristics of all equations that can be solved by roots. But he failed to finish this work because of his early death, and he only solved the first kind of problems. A few years later, Galois took over his work and thoroughly solved the theoretical problem of solvability of algebraic equations by group method, thus establishing the so-called Galois theory.

In the 300 years before the19th century, mathematicians have been busy proving whether there are solutions to equations with one variable and more than four degrees. It's a pity that they either retreat or give up halfway, and no one can untie this knot. 18 18, a Norwegian 16-year-old Albert, after studying a lot of predecessors' information about this problem, firmly said to his teacher, "Let me solve this historical problem, and I can prove whether the equation has been solved more than four times." With confidence, cleverness and diligence, it took him six years to give a satisfactory answer to history: there is no algebraic solution for equations higher than quartic. This is the famous Albert-Ruffini theorem.