Symmetry and Solutions of Polynomial Equations

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Symmetry and Solutions of Polynomial Equations


The search for the solutions of polynomial equations continued throughout the eighteenth century. Within a period of a couple of years around 1770, mathematicians in a number of countries made almost simultaneous advances that led to new ways of approaching the problem. There was disappointment in that none of the advances led to a general solution of the higher-degree equations, but some specific equations could be solved in the aftermath of this work. More importantly, the new approaches provided the means for answering the question about the solvability of polynomial equations in general early in the next century.


An equation is simply an expression in which the two sides are asserted to be equal, which may not involve anything more than numbers. Early on in the development of algebra, the equations of greatest interest were those in which some symbol was used to stand for an unknown quantity whose value was to be determined on the basis of the information furnished by the equation. Linear equations (those involving only the variable to the first power) could be solved without much notation, and it was impressive that Babylonian mathematics was already up to the level of solving quadratic equations (those involving second powers of the variable as well). The description of the solution was usually in terms of words rather than symbols.

That same approach continued even into the sixteenth century with the solution of cubic equations (including third powers of the variable) and quartic equations (allowing fourth powers of the variable). These were solved by the efforts of Italian mathematicians, and the solutions were presented in words. The methods were not scrutinized carefully so long as the answers that were obtained worked in the original equation. Only with the work of François Viète (1540-1603) was there some attempt to apply the rigor of geometry to the techniques of algebra.

Despite Viète's efforts, however, no progress was made in the general question of solving quintic equations (those involving fifth powers of the variable). Towards the end of the eighteenth century Alexandre-Théophile Vandermonde (1735-1796) read a paper before the Académie des Sciences in Paris about a new approach to solving quintic equations and polynomial equations in general. He claimed that every equation in which a power of the variable is set equal to one could be solved by standard means. His work on the subject would be better known if he had been a member of the Academy at the time of delivery (1770), because it would have made it easier for him to publish his claims. As it was, his paper was not published until 1774, and in the meantime Joseph Louis Lagrange (1736-1813), one of the great mathematicians of the century, had published a couple of papers on the subject. Perhaps disappointed, knowing that being the first to publish in the scientific world is of crucial importance, Vandermonde did not pursue his researches into the solution of polynomial equations any further.

Another mathematician whose work anticipated some of Lagrange's was Edward Waring (1736-1798). In his case, the disadvantage that bedeviled his making further progress was the backward state of mathematics in England during his lifetime. As a result of a conscious decision by English mathematicians to follow the practices of Sir Isaac Newton (1642-1727) rather than the leading scholars in Europe, English mathematics drifted away from the advances made in the rest of the world. Waring's work received high praise from Lagrange and others, but he was not in an environment where progress was easy.

Lagrange, by contrast, spent time in the mathematical centers of Europe with the result that his work could have rapid dissemination. His approach to polynomial equations was partly historical, as he stepped back to examine how the Italian mathematicians had managed to come up with their solutions for cubic and quartic equations. On the basis of that analysis, he managed to formulate a research program for how to solve quintic (and higher-degree) equations. As it turned out, even though he was unable to come up with anything by way of a general solution to the quintic equation, his reevaluation of his predecessors provided essential elements for the conclusion at which Evariste Galois (1811-1832) arrived, namely, that the quintic equation could not be solved by traditional means.


The idea that Lagrange brought to the solution of polynomial equations was that of symmetry. Instead of simply looking for solutions to the original equation, he looked for other relationships that the solutions must satisfy in the hope that they would be simpler than the original equation. So long as it was still possible to recover the original solutions from those of the new auxiliary equations, the method would work to supply those original solutions.

If one looks at a quadratic equation (with the coefficient of the squared term equal to one), then the sum of the solutions of the equation is equal to the negative of the coefficient of the first-order term. The product of the solutions is equal to the constant term. This approach would supply an alternative to the quadratic formula for solving a quadratic equation. Similar strategies give auxiliary equations for solving cubic and quartic equations, as the cubic equation can be reduced to a quadratic auxiliary and the quartic to a cubic auxiliary.

The problem with applying Lagrange's strategy to fifth-degree equations became evident in the work of Gian Francesco Malfatti (1731-1807). In a paper in 1770 he showed that if one tries to find an auxiliary equation for a quintic, the result will be of the sixth degree. Since the exponent of the variable in this auxiliary is higher than it was in the original, it has only made the problem worse. When sixth-degree equations showed up in solving the cubic equation, they could be treated as quadratics. The ones that arose from working with the quintic could not be reduced in that way.

Lagrange's work served as the basis for a new discipline of mathematics called group theory. This can be regarded as the study of symmetries, arising out of Lagrange's attempts to find symmetric functions of the solutions of an equation. A function is symmetric if some interchange of values of the variables representing the solutions does not change the value of the function. For example, x + y retains the same value if x and y are interchanged.

Group theory proved to be the basis for the proof that there was no formula for solving all quintic equations as there had been for the lower powers of the variable. This did not mean that individual quintic equations could not be solved, as some of them had been handled centuries earlier. What the proof did guarantee was that any method of solution that claimed to tackle all quintic equations had to go beyond the standard algebraic operations to which attention had previously been limited. The entire branch of mathematics known as Galois theory is devoted to the application of group theory (based on symmetry) to the study of polynomial equations and their solutions.

At the end of the eighteenth century it might have seemed as though no progress had been made on the long-standing problem of solving polynomial equations of degree higher than four. In fact, however, the efforts of Lagrange and others had changed the problem into one that could be approached by the study of symmetry. Lagrange's clarification of what constituted a method of solution for a polynomial equation made it possible for different disciplines within mathematics to be applied to finding those solutions.


Further Reading

Katz, Victor J. A History of Mathematics: An Introduction. New York: HarperCollins, 1993.

Nov ́y, Lubos ̌. Origins of Modern Algebra. Prague: Academia Publishing House, 1983.

Van der Waerden, B.L. A History of Algebra. Berlin: Springer-Verlag, 1985.