Theory of Equations Chapter 4

1. Theory of Equations Chapter 4

2. Belief in algebraic solvability shaken In the 17th century, the belief in the algebraic solvability (in radicals) of all polynomial equations seems to have been in little dispute. The question of solvability was not an issue when the prominent mathematicians such as LEIBNIZ and TSCHIRNHAUS searched for a general solution. Midway through the 18th century the problem had taken a slight turn when EULER in 1732 proposed to investigate the hypothesis that the roots of the general nth degree equation could be written as a sum of n − 1 root extractions of degree n − 1. Although he advanced this as a hypothesis and his search for definite proof was in vain, he based his 1749 “proof” of the Fundamental Theorem of Algebra on the belief that any polynomial equation could be reduced to pure equations1, i.e. radicals .

3. Towards the end of the 18th century, the outspoken beliefs of the most prominent mathematicians had changed, though. Mathematicians with a keen interest in the subject started to suspect that the reduction to pure equations was beyond not only the capacities of themselves but without grasp reach of their existing tools. And at the turn of the century one of the most influential mathematicians, GAUSS, declared the reduction to be outright impossible. The belief in the algebraic solvability of general equations did not vanish completely with GAUSS’ proclamation of its impossibility.

4. On the British Isles, WARING had recognized patterns which led him to the known solutions of low degree equations. With an inclination towards analogies, he suggested that these could be extended to give solutions to all equations, but that the amount of involved computations would explode beyond anything practical.

5. Thus, WARING’s position concerning the possibility of solving higher degree equations algebraically was ambivalent. It remains unclear exactly what it meant to him that he could construct solutions but that the effort required would be infinite.

6. Both WARING and LAGRANGE believed by 1770 that the theories which they had advanced carried in them the solution of the general equations. However, they both acknowledged that the amount of work required to apply these theories to the quinticequation was beyond their own limitations. Before the end of the century, even more radical opinions were to be voiced in print.

7. In the introduction to his first proof (published 1799 but constructed two years earlier) of the Fundamental Theorem of Algebra, GAUSS gave detailed discussions and criticisms of previously attempted proofs. In EULER’s attempt dating back to 1749, GAUSS found the implicit assumption that any polynomial equation could be solved by radicals.

8. “In a few words: It is without sufficient reason assumed that the solution of any equation can be reduced to the resolution of pure equations. Perhaps it would not be too difficult to prove the impossibility for the fifth degree with all rigor; I will communicate my investigations on this subject on another occasion. At this place, it suffices to emphasize that the general solution of equations, in this sense, remains very doubtful, and consequently that any proof whose entire strength depends on this assumption in the current state of affairs has no weight.”

9. In 1799, GAUSS was content to advance his suspicion that the algebraic solution of general equations was not rigorously founded in order to scrutinize EULER’s proof of the Fundamental Theorem of Algebra. Two years later in his influential Disquisitiones(1801), he addressed the problem again in connection with the cyclotomic equations. Possibly alluding to LAGRANGE’s “very great computational work” GAUSS described the solution of higher degree equations not merely beyond the existing tools of analysis but outright impossible.

10. While GAUSS was voicing his opinion on the unsolvabilityof higher degree equations in Latin from his position in G¨ottingen, the foundations under the solvability of the quinticwere shaken even more radically by an Italian. PAOLO RUFFINI had published his first proof of the impossibility of solving the quintic in 1799, the same year GAUSS had first uttered his doubt about its possibility.

11. But where GAUSS had only alluded to a proof without communicating it, RUFFINI had taken the step of publishing his arguments. Together with the early works of A.-L. CAUCHY on the theory of permutations, these make up the final prerequisite of the rigorous breakthrough in the theory of equations.

12. Chapter 5 P. RUFFINI and A.-L. CAUCHY

13. In the 1820s, the search for algebraic solution formulae for equations of higher degree was proved to be in vain when NIELS HENRIK ABEL demonstrated the algebraic unsolvability of the quintic. However, ABEL was not the first to claim the unsolvability; more than 25 years before him, the Italian PAOLO RUFFINI had published his investigations which led him to the same conclusion. RUFFINI’s works were not widely known, and during his investigations ABEL was unaware of their existence. Instead, ABEL based his investigations on the analysis by LAGRANGE and works of CAUCHY on the theory of permutations.

14. The importance of RUFFINI and CAUCHY in the development leading to ABEL’s work on the theory of equations is two-fold. First of all, these men smoothed the transition from the beliefs described in the previous chapter to the rigorous knowledge of the unsolvability of the quintic. Secondly, their investigations took the still young theory of permutations to a more advanced level; and in doing so, they provided an important characterization of the number of values a rational function can obtain under permutations of its arguments.

15. Although GAUSS had proclaimed his belief that the unsolvability of the quintic might not be difficult to prove with all rigor, the Italian P. RUFFINI, in 1799, was the first mathematician to state the unsolvability as a result and attempt to provide it with a proof. RUFFINI’s style of presentation was long, cumbersome, and at times not free of errors; and his initial proof was met with immediate criticism for all these reasons.

16. But convinced of the result and his proof, RUFFINI kept elaborating and clarifying his theory in print for the next 20 years, producing a total of five different versions of the proof. The proofs were published in Italian as monographs in Bologna and in the mathematical memoirs of the Societ`aItalianaelleScienze, Modena. Although published and distributed, the impact of RUFFINI’s work was limited; among the few non-Italians to take a viewpoint on RUFFINI’s work was CAUCHY which he elaborated on numerous occasions, and his final proof (1813).

17. The writings of RUFFINI were thoroughly inspired by LAGRANGE’s analysis of the solvability of equations (1770–1771). Its ideas, concepts, and notation permeate RUFFINI’s works; and on numerous occasions RUFFINI openly acknowledged his debt to LAGRANGE. As LAGRANGE had done, RUFFINI studied equations of low degrees in order to establish patterns subject of generalization. Prior to applying his analysis to the fifth degree equation, RUFFINI propounded the corner stone of his investigation.

18. Central to his line of argument was his classification of permutations. Founded in LAGRANGE’s studies of the behavior of functions when their arguments were permuted, RUFFINI set out to classify all such permutations of arguments which left the function (formally) unaltered. RUFFINI’s concept of permutation (Italian: “permutazione”) differed from the modern one, and can most easily be understood if translated into the modern concept introduced by CAUCHY in the 1840s of systems of conjugate substitutions4. Thus, a permutation for RUFFINI corresponded to a collection of inter-changements(transitions from one arrangement to another) which left the given function formally unaltered.

19. RUFFINI divided his permutations into simple ones which were generated by iterations (i.e. powers) of a single interchangement6 and composite ones generated by more than one inter-changement. His simple permutations consisting of powers of a single inter-changementwere subdivided into two types distinguishing the case in which the single interchangement consisted of a single cycle from the case in which it was the product of more than one cycle.

20. RUFFINI’s composite permutations were subsequently subdivided into three types. A permutation was said to be of the first type if two arrangements existed which were not related by an inter-changementfrom the permutation. Translated into the modern terminology of permutation groups, this type corresponds to intransitive groups. RUFFINI defined the second type to contain all permutations which did not belong to the first type and for which there existed some set of roots S. Such transitive groups were later termed imprimitive. The last type consisted of any permutation not belonging to any of the previous types, and thus corresponds to primitive groups.

21. Building on this classification of all permutations into the five types RUFFINI introduced his other key concept of degree of equivalence (Italian: “gradodiuguaglianza”) of a given function f of the n roots of an equation as the number of different permutations not altering the formal value of f. Denoting the degree of equivalence by p, RUFFINI stated the result of LAGRANGE that p must divide n!.

22. In order to prove the impossibility of solving the quintic algebraically, RUFFINI assumed without proof that any radical occurring in a supposed solution would be rationally expressible in the roots of the equation. He never verified this hypothesis, and it is considered one of the greatest advances of ABEL’s proof over RUFFINI’s that he independently focused on the same hypothesis and provided it with a proof.

23. The proof which RUFFINI gave for the unsolvability of the quintic was thus based on three central parts: 1. The classification of permutations into types. 2. A demonstration, that no function of the five roots of the general quintic could have 3, 4, or 8 values under permutations of the roots. 3. A study of the two inner-most (first) radical extractions of a supposed solution to the quintic, in which the result of (2) was used to reach a contraction.

24. The mere extent of the classification and the caution necessary to include all cases combined with RUFFINI’s intellectual debt to LAGRANGE may serve to view RUFFINI’s work as filling in some of the “infinite labor” described by WARING and LAGRANGE in expressing their doubts about the solvability of higher degree equations. But at the outcome of RUFFINI’s investigations (and probably also at the outset), he obtained the complete reverse result: that the solution of the quintic was impossible.

25. One of RUFFINI’s friends and critical readers, named PIETRO ABBATI (1768–1842), gave several improvements of RUFFINI’s initial proof. The most important one was that he replaced the laborious arguments based on thorough consideration of particular cases by arguments of a more general character10. These more general arguments greatly simplified RUFFINI’s proofs that no function of the five roots of the quintic could have 3, 4, or 8 different values. ABBATI was convinced of the validity of RUFFINI’s result but wanted to simplify its proof, and RUFFINI incorporated his improvements into subsequent proofs, from 1802 and henceforth.

26. Others, however, were not so convinced of the general validity of RUFFINI’s results. Mathematicians belonging to the “old generation” were somewhat stunned by the non-constructive nature of the proofs, which they described as “vagueness”. For instance, the mathematician GIAN FRANCESCO MALFATTI (1731–1807) severely criticized RUFFINI’s result since it contradicted a general solution which he, himself, previously had given. RUFFINI responded by another publication of a version of his proof answering to MALFATTI’s criticism; but before the discussion advanced further, MALFATTI died.

27. In his fifth, and final, publication of his unsolvability theorem (1813), RUFFINI recapitulated important parts of LAGRANGE’s theory, in which he emphasized the distinction between numerical and formal equality, before giving the refined version of his proof. In (1845), PIERRE LAURENT WANTZEL (1814–1848) gave a fusion argument incorporating the permutation theoretic arguments of RUFFINI’s final proof into the setting of ABEL’s proof. RUFFINI corresponded with AUGUSTIN-LOUIS CAUCHY, who in 1816 was a promising young Parisian ingenieur. CAUCHY praised RUFFINI’s research on the number of values which a function could acquire when its arguments were permuted, a topic CAUCHY, himself, had investigated in an treatise published the year before (1815) with due reference to RUFFINI.

28. Following this exchange of letters CAUCHY wrote RUFFINI another letter in September 1821, in which he acknowledged RUFFINI’s progress in the important field of solvability of algebraic equations: “I must admit that I am anxious to justify myself in your eyes on a point which can easily be clarified. Your memoir on the general solution of equations is a work which has always appeared to me to deserve to keep the attention of geometers. In my opinion, it completely demonstrates the algebraic unsolvability of the general equations of degrees above the fourth. The reason that I had not lectured on it [the unsolvability] in my course in analysis

29. And it must be said that these courses are meant for students at the ´ EcoleRoyale Polytechnique, is that I would have deviated too much from the topics set forth in the curriculum of the ´ Ecole. At least by 1821, the validity of RUFFINI’s claim that the general quintic could not be solved by radicals was propounded, not only by a somewhat obscure Italian mathematician, but also one of the most promising and ambitious French mathematicians of the early 19th century. However, it should take further publications, notably by the young ABEL, before this validity would be accepted by the broad international community of mathematicians.

30. In November of 1812, CAUCHY handed in a memoir on symmetric functions to the First Class of the Institut de France which was published three years later as two separate papers, (1815a) and (1815b), in the Journal d’E´colePolytechnique. The first of the two papers (1815a) is of special interest in the history of solvability of polynomial equations. Although CAUCHY’s issue was not the solvability question, his paper was to become extremely important for subsequent research. It was primarily concerned with a more general version of RUFFINI’s result that no function of five quantities could have three or four different values when its arguments were permuted.

31. Before going into this particular result, however, CAUCHY devised the terminology and notation which he was going to use. Precisely in formulating exact and useful notation and terminology CAUCHY advanced well beyond his predecessors and laid the foundations upon which the 19th century theory of permutations would later build.

32. With CAUCHY the term “permutation” came to mean an arrangement of indices, thereby replacing the “arrangements” of which RUFFINI spoke. A “substitution” was subsequently defined to be a transition from one permutation to another (which is the modern meaning of “permutation”)

33. - The End – Happy Valentines Day!