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Theory of Equations Chapter 6 Algebraic Unsolvability of the Quintic : limiting the class of solvable equations.
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Theory of Equations Chapter 6 Algebraic Unsolvabilityof the Quintic: limiting the class of solvable equations
In spite of the efforts of RUFFINI and GAUSS, the search for algebraic solution formulae for the quintic remained an attractive problem to a generation of young and aspiring mathematicians. In Norway, ABEL thought he had solved it, but soon realized that he had been misled. In Germany, CARL GUSTAV JACOB JACOBI (1804–1851) worked on the problem, and in France GALOIS, too, thought he had found a solution, only to soon be disappointed. All of them attacked the problem while they still attended pre-university schools. The problem’s easy formulation and yet century-long history, and a general belief that its solution should be possible and not too difficult, made it appear as a good opening into doing creative mathematics.
Inspired by the stimulation of his new, and young, mathematics teacher B. M. HOLMBOE, ABEL studied the masters and began to engage in creative mathematics of his own. In 1821, he thought he had produced a solution to the general fifth degree equation. In the incipient intellectual atmosphere of Christiania, few authorities capable of determining the validity of ABEL’s reasoning could be found. But more importantly, the scientific milieu of Norway was still without a means of publication of technical mathematical results deserving international recognition.
For these reasons, professor CHRISTOPHER HANSTEEN (1784–1873) sent ABEL’s manuscript to professor FERDINAND DEGEN (1766–1825)3 in Copenhagen for evaluation and possibly publication in the transactions of the Royal Danish Academy of Sciences and Letters. The accompanying letter, which HANSTEEN must have written, and the paper are no longer preserved. Our only primary source of information is the letter which DEGEN wrote back to HANSTEEN, in which he asked for an elaborated version of the argument and an application to a specific numerical example.
We have no indication that ABEL ever produced an elaborated deduction; apparently the numerical examples worked their part—as the probes of truth—as DEGEN had suggested and led ABEL to a radically new insight. In 1824, he published, at his own expense, a short work (Abel 1824) in French entitled M´emoiresur les ´equations alg´ebriquesoul’on d´emontre l’impossibilit´e de la r´esolution de l’´equation g´en´erale du cinqui`emedegr´e. As ABEL announced in the title, it demonstrated the impossibility of solving the general equation of the fifth degree. ABEL intended the memoir to be his best self-introductionnon his planned tour of the Continent.
Since he had had to pay for the publication himself, it only covered six pages, and his style of presentation suffered accordingly. In numerous points he was unclear or left advanced arguments out. But when ABEL came into contact with A. L. CRELLE in Berlin, he found himself in a position to make his discovery available to a broader public. He rewrote the argument elaborating the ideas of the 1824 proof, and had CRELLE translate it into German for publication in the very first issue of Journal f¨urdie reine und angewandteMathematik (Abel 1826a). Through this treatise—and the French review which ABEL wrote of it for BARON DE FERRUSAC’s (1776–1836) Bulletin des sciences math´ematiques, astronomiques, physiques et chimiques (Abel 1826b) the world gradually came to know that a young Norwegian had settled the question of solvability of the general quintic in the negative.
In the opening paragraph of the treatise in CRELLE’s Journal, ABEL described the approach he had taken. In order to answer the question of solvability of equations, he proposed to investigate the forms of all algebraic expressions in order to determine if they could “solve” the equation. Although ABEL throughout spoke of algebraic functions, I use the term algebraic expressions to avoid any untimely inference from the modern conception of functions as mappings. The algebraic expressions which ABEL considered were algebraic combinations of the coefficients of the given equation, and thus his approach was in line with the one taken earlier by VANDERMONDE.
“As is known, the algebraic equations up to the fourth degree can be solved in general. Equations of higher degrees, however, only in particular cases, and if I am not mistaken, the question: Is it possible to solve equations of higher than the fourth degree in general? has not yet been answered in a satisfactory manner. The present treatise is concerned with this question. To solve an equation algebraically is but to express its roots by algebraic functions of its coefficients.
Therefore, one must first consider the general form of algebraic functions and subsequently investigate whether the given equation can possibly be satisfied by inserting the expression of an algebraic function in place of the unknown quantity. This shift from the trial-and-error based search for solution formulae to a theoretical and general investigation of the class of algebraic expressions marks ABEL’s first break with the traditional approach in the theory of equations.
ABEL investigated the extent to which algebraic expressions could satisfy given polynomial equations and was led to describe necessary conditions. By this choice of focal point, ABEL implicitly introduced a new object, algebraic expression, into the realm of algebra, and the first part of his treatise can be seen as a preliminary study of this object, devised in order to obtain a firm description of it and to prove its first central theorem.
The treatise in CRELLE’s Journal can be divided into four sections reflecting the overall structure of ABEL’s proof. In the first section, ABEL introduced his definition of algebraic functions and classified these by their orders and degrees. He used this definition to study the restrictions imposed on the form of algebraic expressions when these were assumed to be solutions to a given solvable equation. In doing so, he proved the result which RUFFINI had failed to see — that any radical contained in a supposed solution would depend rationally on the roots of the equation.
In the second section, ABEL reproduced the elements of CAUCHY’s theory of permutations (1815a) needed for his proof. These included CAUCHY’s notation and the result described above as the CAUCHY-RUFFINI theorem (section 94) demonstrating that no function of the five roots of the general quintic could take on three or four different values under permutations of these roots. The third part contained detailed and highly explicit investigations of functions of five quantities taking on two or five different values under all permutations of the roots. Through an explicit theorem, which linked the number of values under permutations to the degree of the root extraction, ABEL demonstrated that all non-symmetric rational functions of five quantities could be reduced to two basic forms.
Finally, these preliminary sections were combined to provide ABEL’s impossibility proof by reducing each of a number of cases ad absurdum. Throughout, ABEL’s approach to the question of solvability of the quintic was based on counting the number of values which a rational function took when its arguments were permuted. Thus, he clearly worked in the tradition initiated by LAGRANGE, and it is remarkable that no reference to or even mention of LAGRANGE was ever made in ABEL’s published works on the theory of equations.
The objects which ABEL called algebraic functions—and which I term algebraic expressions were finite combinations of constant and variable quantities obtained by basic arithmetical operations. If the operations included only addition and multiplication, the expression was said to be entire; if, furthermore, division was involved, it was called rational; and if, additionally, root extractions of prime degree were allowed, the expression was denoted an algebraic expression. Subtraction and extraction of roots of composite degree were explicitly considered to be contained in the above operations. In the subsequent classification, ABEL benefitted from the simplicity introduced by this minimal definition.
The implicit purpose of ABEL’s investigations of algebraic expressions was to obtain an important auxiliary theorem for his impossibility proof. Based on a definition which introduced algebraic expressions as objects, ABEL derived a standard form for these objects. Applying it to algebraic expressions which satisfied a given equation, he found that these could always be given a form in which all occurring parts depended rationally on the roots of the equation. In his effort to obtain a classification of algebraic expressions, ABEL introduced a hierarchy based on the concepts of order and degree. These concepts introduced a structure in the class of algebraic expressions allowing ordering and induction to be carried out.
In dealing with the proof which ABEL gave of his auxiliary theorem, we are introduced to two other concepts which are even more fundamental to his theory of algebraic solvability. These are the Euclidean division algorithm and the concept of irreducibility. The proof is presented in quite some detail to demonstrate how ABEL made use of these concepts. They were to become even more important in his unpublished general theory of solvability
When ABEL published his proofs of the impossibility result (1824) and (1826a), he was allegedly unaware of the proofs of RUFFINI. Since questions of priority have often been a motivation for writing (and rewriting) the history of mathematics, this independence of results is noticed by most biographers of ABEL. It is my firm conviction based on the mathematical contents of his proof that ABEL devised his proof independently of RUFFINI. However, the primary sources of information on ABEL’s independence of RUFFINI are limited. The only mention of RUFFINI made by ABEL is in his notebook entry on the theory of solvability, in the introduction to which he described RUFFINI’s proof:
“The first person, and if I am not mistaken, the only one prior to me, who has tried to prove the impossibility of the algebraic solution of the general equations, is the geometer Ruffini; but his memoir is so complicated that it is very difficult to judge the validity of his reasoning. It seems to me that his reasoning is not always satisfying. I think that the proof I gave in the first issue of this journal [CRELLE’s Journal] leaves nothing to be desired as to rigor, but it does not have all the simplicity of which it is susceptible. I have reached another proof based on the same principles, but more simple, in trying to solve a more general problem.”
The notebook has been dated to 182826 by PETER LUDVIG MEJDELL SYLOW (1832– 1918) a date which implies that once back in Christiania ABEL disclosed his knowledge of RUFFINI. It is most likely that ABEL learned about RUFFINI during his European tour, and two instances are of main importance. During his stay in Vienna in April and May 1826, ABEL became acquainted with the local astronomers KARL LUDWIG VON LITTROW (1811–1877) and ADAM, FREIHERR VON BURG (1797–1882).
In the first volume of their journal Zeitschriftf¨urPhysik und Mathematik, which occurred while ABEL was in town, an anonymous paper on the theory of equations (Anonymous 1826) was published. The author, who was inspired by ABEL’s proof and praised it highly, reviewed RUFFINI’s proof. Therefore it is not unlikely that ABEL learned of RUFFINI’s proof from his Viennese connections. Once in Paris, ABEL took on the duty of writing unsigned reviews for FERRUSAC’s Bulletin des sciences math´ematiques, astronomiques, physiques et chimiques of paperspublished in CRELLE’s Journal.
Thus, at two instances in 1826 ABEL had been in close contact with journals, in which his result was linked to that of RUFFINI. A third possible source of information on RUFFINI’s research was, of course, CAUCHY whom ABEL met in Paris without any traceable interaction taking place. Although the primary information on how ABEL came to know of RUFFINI’s proofs is rather sparse, I find further support in the mathematical technicalities for the assumption of independence.
Their differences in notation and approach to permutations, ABEL definition of algebraic expressions and his careful proof of the auxiliary theorems describing them all suggest to me, that ABEL’s deduction was a custom made argument for the impossibility, independent of any earlier such proofs. The common inspiration from LAGRANGE, which both authors admitted, should be evident enough to account for similarities in studying the blend of equations and permutations. At a conceptual level ABEL’s proof that the general quintic could not be solved algebraically was more than just another proof in the body of mathematics.
In denying that the problem of determining the solution to the fifth degree equation which had engaged mathematicians for centuries could be solved, it provided one of the negative results which were only just starting to dominate mathematics. Any result can be formulated as a negative one, but negative in this connection also indicates some degree of counter-intuition. ABEL had demonstrated that any supposed solution to the general quintic carried with it an internal contradiction, and thus the result not only made the belief in algebraic solvability tremble, it completely destroyed it.
In 1826, when ABEL made his proof available to a broader public for the first time, the mathematical world had already seen the first constructions of non-Euclidean geometries implying the impossibility of deducing the fifth postulate as a theorem, but it would still take some time for the full consequences of these to be realized, too. The reaction of the mathematical community to the impossibility proofs in the theory of equations can be divided in three. Some mathematicians, often belonging to the older generation or the laity of mathematics, protested against the result and held both the statement and the proof to be flawed. Others accepted the result, but provided refinements of the proofs and their assumptions.
And yet others not only accepted the results, but saw that the quintic constituted an example of a unsolvable equation, whereby the question of solvability had been isolated. The quintic provided an example of an equation not belonging to the set of algebraically solvable equations. On the other hand GAUSS had demonstrated that infinitely many algebraically solvable equations existed, so the set of algebraically solvable equation did not collapse to a few low degree equations. Therefore the problem of deciding whether a given equation was solvable or not emerged as an interesting project for research.
