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Chapter 2 Aspects of the theory of equations from the 16th to the early 19th century

Chapter 2 Aspects of the theory of equations from the 16th to the early 19th century.

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Chapter 2 Aspects of the theory of equations from the 16th to the early 19th century

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  1. Chapter 2Aspects of the theory of equations fromthe 16th to the early 19th century

  2. To mathematicians of the 17th and 18th centuries algebra, or analysis finitorum as it was commonly called, consisted of the study of formal relations between coefficients and roots, and the solution of polynomial equations. In this chapter I describe the evolution of some of the main problems in the field, and how the most important results available at the dawn of the 19th century came to be. The main question of the present study is that of algebraic solvability. 2.1 The existence of roots When REN´E DU PERRON DESCARTES (1596–1650) in 1637 claimed that any equation of degree n possessed exactly n roots a central problem of algebra was formulated1. His way out was a rather evasive one which consisted of distinguishing the real ones (real meaning “in existence”) from the imaginary ones which were products of human imagination. To DESCARTES the assertation that any equation of degree n had n roots took the form of a general property possessed by all equations and the trick of introducing the imagined2 roots saved him from further argument.

  3. To the next generations of mathematicians the character of the problem changed. Whereas DESCARTES had not dealt with the nature of the imagined roots, they did. Soon the problem of demonstrating that all (imagined) roots of a polynomial equations were complex, i.e. of the form a + b√−1 for real a, b, was raised; and the around the time of GAUSS the theorem acquired the name of the Fundamental Theorem of Algebra. When GOTTFRIED WILHELM LEIBNIZ (1646–1716) doubted that the polynomial x4 + c4 could be split into two real factors of the second degree the validity of the result seemed for a moment in doubt. EULER demonstrated in 1749 (published 1751) that the set of complex numbers was closed under all algebraic and numerous transcendental operations.

  4. Thus, at least by 1751 it would implicitly be known that √i = 1+i/ √ 2, which made LEIBNIZ’s supposed counter-example evaporate. Numerous prominent mathematicians of the 18th century — among them notably JEAN LE ROND D’ALEMBERT (1717–1783), EULER, and LAGRANGE — sought to provide proofs that any polynomial could be split into linear and quadratic factors which would prove that any imagined roots were indeed complex. In the half-century 1799–1849 GAUSS gave a total of four proofs6 which, although belonging to an emerging trend of indirect existence proofs, were considered to be superior in rigor when compared to those of his predecessors. Today these proofs are generally credited with being the first rigorous proofs of this important theorem. The nature of the proofs varied from considerations of infinite series by D’ALEMBERTand essentially topological7 approaches by GAUSS, to formal manipulations of coefficientsand equations by EULER. The proofs borrowed techniques and arguments fromboth algebra and (infinite) analysis, analysis infinitorum. In general, the FundamentalTheorem of Algebra and its proofs were and are more integrated intoanalysis than into algebra.

  5. Characterizing roots The proofs of the Fundamental Theorem of Algebra were largely non constructive existence proofs and other non-constructive results were also pursued. An important subfield of the theory of equations was developed in order to characterize and describe properties of the roots of a given equation from a priori inspections of the equation and without explicitly knowing the roots. LAGRANGE motivated his research to describe properties of the roots of particular equations by the general problems arising from attempts to solve higher degree equations through algebraic expressions (see below)8. The interest of LAGRANGE in numerical equations, i.e. concrete equations in which some dependencies among the coefficients can exist, can be divided into three topics: the nature and number of the roots, limits for the values of these roots, and methods for approximating these. LAGRANGE made use of analytic geometry, function theory, and the Lagrangian calculus—methods belonging to analysis infinitorum — in order to investigate these topics. It was LAGRANGE’s aim to establish a purely analytic foundation—using both algebra and analysis—for the theory of equations.

  6. Positive and negative roots. From the earliest days of Western theory of equations, only positive roots had been considered as in existence and negative numbers were never thought of. Negative numbers and negative roots were considered false or fictuous by GIROLAMO CARDANO (1501-1576), who devised a method of determining false roots of one equation by finding true roots of another10. By the time of DESCARTES, the distinction had been weakened a bit, and he allowed both positive “true” and negative “false” roots of an equation11. These roots were still not, though, on a par and the famous Rule of Descartes, which generalized CARDANO’s result, provided a tool for establishing bounds on the number of positive or negative roots possessed by a given equation by counting the changes of sign in the sequence of coefficients. When seen as an example of expressing properties of the essentially unknown roots of an equation, this initiated a research branch for the following centuries. Mathematicians of the 18th and 19th centuries sought to prove the Rule of Descartes and in doing so JEAN PAUL DE GUA DE MALVES (1712-1786)generalized it to also determine bounds on the possible numbers of complex roots12. Dating back to SIMON STEVIN’s (1548–1620) approximative solutions of third degree equations, methods of obtaining bounds for the roots from the coefficients of the equation were known. Combined with quite simple transformations such results were used by DESCARTES to obtain from a given equation with both positive and negative roots another one in which all the roots were positive through a translation of the form y = x − a. In the 19th century, CAUCHY developed his residual theory of functions and used it to determine the number of roots of a polynomial contained in a given bounded region of the complex plane14. In obtaining the topological descriptions of the unknown roots of equations, analytical methods were put to great use in the theory of equations.

  7. Elementary symmetric relations. A different but in connection with the present study more important example of a priori properties of the roots of an equation was conceived of by men as CARDANO, FRANC¸ OIS VI`E TE (1540-1603), and ISAAC NEWTON (1642–1727) in the 16th and 17th centuries. From inspection of equations of low degree they obtained (generally by analogy and without general proofs) the dependency of the coefficients of the equationxn + an−1xn−1 + an−2xn−2 + . . . + a1x + a0 = 0 on the roots x1, . . . , xn given by an−1 = −(x1 + . . . + xn) an−2 = x1x2 + . . . + xn−1xn...a1 = ±(x1x2 · · · xn−1 + . . . + x2x3 · · · xn) a0 = x1x2 · · · xn.

  8. These equations established the Elementary symmetric relations between the roots and the coefficients of an equation. When proofs of these relations emerged, they were obtained through formal manipulations and were, thus, firmly within the established algebraic style. The relations were to become a central tool in the theory of equations, especially after they had been demonstrated to be the basic, or elementary, ones on which all other symmetric functions of the roots depended rationally.

  9. 2.3 Resolvent equations From the multitude of possible questions concerned with describing the unknown roots one is particularly linked to the question of solving equations algebraically. It is concerned with the form in which the roots can be written and is thus a first step in the direction of solvability questions. The general approach taken in solving equations of degrees 2, 3 or 4 had since the first attempts been to reduce their solution to the solution of equations of lower degree. The example of the third degree equation solved by SCIPIONE FERRO (1465-1526) around 1539, by NICCOL`O TARTAGLIA (1499/1500–1557) in 1539, and by CARDANO, who published the solution in 1545, might be illustrative.

  10. Chapter 3Study the Masters

  11. J. L. LAGRANGE WhenLAGRANGE in 1770–1771 hadhisR´eflexions sur la r´esolutionalg´ebriquedes´equations published in the M´emoires of the Berlin Academy, he was a well established mathematician held in high esteem. The R´eflexions was a thorough summary of the nature of solutions to algebraic equations which had been uncovered until then. Much as EULER and VANDERMONDE had done, LAGRANGE investigated the known solutions of equations of low degrees hoping to discover a pattern feasible to generalizations to higher degree equations. Where EULER had sought to extend a particular algebraic form of the roots, and VANDERMONDE had tried to generalize the algebraic functions of the elementary symmetric functions, LAGRANGE’s innovation was to study the number of values which functions of the coefficients could obtain under permutations of the roots of the equation. Although he exclusively studied the values of the functions under permutations, his results marked a first step in the emerging independent theory of permutations. In turn, this permutation theory was soon, through its central role in GALOIS’s theory of algebraic solvability, incorporated in the abstract theory of groups which grew out of 19th and 20th century abstraction. The work R´eflexionssur la r´esolutionalg´ebrique des ´equations (1770–1771) was divided into four parts reflecting the structure of LAGRANGE’s investigation. 1. “On the solution of equations of the third degree” (Lagrange 1770–1771, 207–254)

  12. 2. “On the solution of equations of the fourth degree” (ibid. 254–304) 3. “On the solution of equations of the fifth and higher degrees” (ibid. 305–355) 4. “Conclusion of the preceding reflections with some general remarks concerning the transformation of equations and their reduction to a lower degree” (ibid. 355–421) Of these the latter part is of particular interest to the following discussion. It concerned providing a link between the number of values a function could obtain under permutations and the degree of the associated resolvent equation. Most accounts of LAGRANGE’s contribution in the theory of equations emphasize the 100th section dealing with the rational dependence of semblablesfonctions, a topic which became central after the introduction of GALOIS theory11. However, as this story mainly concerns the theory of equations prior to GALOIS’ theory—and in particular ABEL’s contributions—the focus will be on other sections.

  13. Formal values of functions The central innovation of LAGRANGE was the idea of studying the number of formally different values which a function would obtain when its arguments were permuted in all possible ways. Before going into the useful results which he obtained through this approach, a closer look at his ideas about formal values and his investigations leading towards permutation theory is worthwhile. Central to LAGRANGE’s treatment of the general equations of all degrees was his concept of formal functional equality. He considered two rational functions equal only when they were given by the same algebraic formulae, whereby xy and yx were considered equal because both multiplication (and addition) were implicitly assumed to be commutative. Denoting the roots of the general μth degree equation by x1, . . . , xμ, LAGRANGE considered the roots as independent, meaning that x1 was never equal to x2. In the background of this can be seen the 18th century conception which did not see the polynomial on the left hand side of the equation as a functional mapping but as an expression combined of various symbols: variables and constants, known and unknown. LAGRANGE was not particularly explicit about this notion of formal equality which occurs throughout his investigations, but in article 103 he wrote that, “it is only a matter of the form of these values and not their absolute [numerical] quantities.” The emphasis on formal values was lifted when GALOIS saw that in order to address special equations, in which the coefficients were not completely independent, he had to consider the numerical equality of the symbols in place of LAGRANGE’s formal equality.

  14. The emergence of permutation theory Another of LAGRANGE’s new ideas was the introduction of symbols denoting the roots which enabled him to compute directly with them14. But more important was the way in which he focused his attention on the action of permutations on formal expressions in the roots. LAGRANGE set up a system of notation in which f [(x’) (x’’) (x’’’)] meant that the function f was (formally) altered by any (non-identity) permutation of x’, x’’, x’’’. If the function remained unaltered when x’ and x’’ were interchanged, LAGRANGE wrote it as f [(x’, x’’) (x’’’)] . And if the function was symmetric (i.e. formally invariant under all permutations of x’, x’’, x’’’), he wrote f [(x’, x’’, x’’’)] . With this notation and his concept of formal equality LAGRANGE derived far-reaching results on the number of (formally) different values which rational functions could assume under all permutations of the roots. With the hindsight that the set of permutations form an example of an abstract group, a permutation group, LAGRANGE was certainly involved in the early evolution of permutation group theory. As we shall see in the following section, he was led by this approach to Lagrange’s Theorem, which in modern terminology expresses that the order of a subgroup divides the order of the group. However, since LAGRANGE dealt with the actions of permutations on rational functions, he was conceptually still quite far from the concept of groups. LAGRANGE’s contribution to the later field of group theory laid in providing the link between the theory of equations and permutations which in turn led to the study of permutation groups from which (in conjunction with other sources) the abstract group concept was distilled. But more importantly, LAGRANGE’s idea of introducing permutations into the theory of equations provided subsequent generations with a powerful tool.

  15. C. F. GAUSS Thirty years after LAGRANGE’s creative studies on known solutions to low degree equations, and in particular properties of rational functions under permutations of their arguments, another great master published a work of profound influence on early 19th century mathematics. From his position in G¨ottingen, C. F. GAUSS was located at a physical distance from the emerging centers of mathematical research in Paris and Berlin. By 1801, the Parisian mathematicians had for some time been publishing their results in French and, within a generation, the German mathematicians would also be writing in their paternal language, at least for publications intended for AUGUST LEOPOLD CRELLE’s (1780–1855) Journal f¨ur die reine und angewandte Mathematik. But when GAUSS published his Disquisitionesarithmeticae (1801), it was written in Latin and published as a monograph as was still customary to his generation of German scholars. The book was divided into seven sections, although allusions and references were made to an eighth section which GAUSS was never to complete for publication. The main part was concerned with the theory of congruences, the theory of forms, and related number theoretic investigations. Together, these topics provided a new foundation, emphasis, and disciplinary independence—as well as a wealth of results for 19th century number theorists in particular GUSTAV PETER LEJEUNE DIRICHLET (1805–1859)— to elaborate. In dealing with the classification of forms, GAUSS made use of “implicit group theory”26 but the abstract concept of groups was almost as far beyond GAUSS as it had been beyond LAGRANGE.

  16. One of the new tools applied by GAUSS in the theory of congruences was that of primitive roots. In the articles 52–57, GAUSS gave his exposition of EULER’s treatment of primitive roots. A primitive root k of modulus μ is an integer 1 < k < μ such that the set of remainders of its powers k1, k2, . . . , kμ−1 modulo μ coincides with the set {1, 2, . . . , μ − 1}, possibly in a different order. A central result obtained was the existence of p − 1 different primitive roots of modulus p if p were assumed to be prime. The division problem for the circle In the seventh section of his Disquisitionesarithmeticae(1801), GAUSS turned his investigations toward the equations defining the division of the periphery of the circle into equal parts. He was interested in the ruler and compass constructibility of regular polygons and was therefore led to study in details how, i.e. by the extraction of which roots, the binomial equations of the form xn− 1 = 0 could be solved algebraically. If the roots of this equation could be constructed by ruler and compass, then so could the regular p-gon. It is evident from GAUSS’ mathematical diary that this problem had occupied him from a very early stage in his mathematical career and had been the deciding factor in his choice of mathematics over classical philology. The very first entry in his mathematical progress diary from 1796 read: “[1] The principles upon which the division of the circle depend, and geometrical divisibility of the same into seventeen parts, etc. [1796] March 30 Brunswick.” (Gray 1984, 106)

  17. In his introductory remarks of the seventh section, GAUSS noticed that the approach which had led him to the division of the circle could equally well be applied to the division of other transcendental curves of which he gave the lemniscate as an example. “The principles of the theory which we are going to explain actually extend much farther than we will indicate. For they can be applied not only to circular functions but just as well to other transcendental functions, e.g. to those which depend on the integral R [1/√(1 − x4)] dx and also to various types of congruences.” (Gauss 1986, 407)30 However, as he was preparing a treatise on these topics GAUSS had chosen to leave this extension out of the Disquisitiones. GAUSS never wrote the promised treatise, and after ABEL had published his first work on elliptic functions (Abel 1827) culminating in the division of the lemniscate, GAUSS gave him credit for carrying these investigations into print.

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