Impasse with radical methods Algebra Mathematic
A major breakthrough in the algebraic solution of higher-degree equations was achieved by the Italian-French mathematician Joseph-Louis Lagrange in 1770. Rather than trying to find a general solution for quintic equations directly, Lagrange attempted to clarify first why all attempts to do so had failed by investigating the known solutions of third- and fourth-degree equations. In particular, he noticed how certain algebraic expressions connected with those solutions remained invariant when the coefficients of the equations were permuted (exchanged) with one another. Lagrange was certain that a deeper analysis of this invariance would provide the key to extending existing solutions to higher-degree equations.
Using ideas developed by Lagrange, in 1799 the Italian mathematician Paolo Ruffini was the first to assert the impossibility of obtaining a radical solution for general equations beyond the fourth degree. He adumbrated in his work the notion of a group of permutations of the roots of an equation and worked out some basic properties. Ruffini’s proofs, however, contained several significant gaps.
Between 1796 and 1801, in the framework of his seminal number-theoretical investigations, Gauss systematically dealt with cyclotomic equations: xp − 1 = 0 (p > 2 and prime). Although his new methods did not solve the general case, Gauss did demonstrate solutions for these particular higher-degree equations.
In 1824 the Norwegian mathematician Niels Henrik Abel provided the first valid proof of the impossibility of obtaining radical solutions for general equations beyond the fourth degree. However, this did not end polynomial research; rather, it opened an entirely new field of research since, as Gauss’s example showed, some equations were indeed solvable. In 1828 Abel suggested two main points for research in this regard: to find all equations of a given degree solvable by radicals, and to decide if a given equation can be solved by radicals. His early death in complete poverty, two days before receiving an announcement that he had been appointed professor in Berlin, prevented Abel from undertaking this program.
A major breakthrough in the algebraic solution of higher-degree equations was achieved by the Italian-French mathematician Joseph-Louis Lagrange in 1770. Rather than trying to find a general solution for quintic equations directly, Lagrange attempted to clarify first why all attempts to do so had failed by investigating the known solutions of third- and fourth-degree equations. In particular, he noticed how certain algebraic expressions connected with those solutions remained invariant when the coefficients of the equations were permuted (exchanged) with one another. Lagrange was certain that a deeper analysis of this invariance would provide the key to extending existing solutions to higher-degree equations.
Using ideas developed by Lagrange, in 1799 the Italian mathematician Paolo Ruffini was the first to assert the impossibility of obtaining a radical solution for general equations beyond the fourth degree. He adumbrated in his work the notion of a group of permutations of the roots of an equation and worked out some basic properties. Ruffini’s proofs, however, contained several significant gaps.
Between 1796 and 1801, in the framework of his seminal number-theoretical investigations, Gauss systematically dealt with cyclotomic equations: xp − 1 = 0 (p > 2 and prime). Although his new methods did not solve the general case, Gauss did demonstrate solutions for these particular higher-degree equations.
In 1824 the Norwegian mathematician Niels Henrik Abel provided the first valid proof of the impossibility of obtaining radical solutions for general equations beyond the fourth degree. However, this did not end polynomial research; rather, it opened an entirely new field of research since, as Gauss’s example showed, some equations were indeed solvable. In 1828 Abel suggested two main points for research in this regard: to find all equations of a given degree solvable by radicals, and to decide if a given equation can be solved by radicals. His early death in complete poverty, two days before receiving an announcement that he had been appointed professor in Berlin, prevented Abel from undertaking this program.
