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Analytic Solutions to Algebraic Equations

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Analytic Solutions to Algebraic Equations Mathematics Tomas Johansson LiTH{MAT{Ex{98{13 Supervisor: Bengt Ove Turesson Examiner: Bengt Ove Turesson LinkÄoping 1998-06-12 CONTENTS 1 Contents 1 Introduction 3 2 Quadratic, cubic, and quartic equations 5 2.1 History . 5 2.2 Solving third and fourth degree equations . 6 2.3 Tschirnhaus transformations . 9 3 Solvability of algebraic equations 13 3.1 History of the quintic . 13 3.2 Galois theory . 16 3.3 Solvable groups . 20 3.4 Solvability by radicals . 22 4 Elliptic functions 25 4.1 General theory . 25 4.2 The Weierstrass } function . 29 4.3 Solving the quintic . 32 4.4 Theta functions . 35 4.5 History of elliptic functions . 37 5 Hermite's and Gordan's solutions to the quintic 39 5.1 Hermite's solution to the quintic . 39 5.2 Gordan's solutions to the quintic . 40 6 Kiepert's algorithm for the quintic 43 6.1 An algorithm for solving the quintic . 43 6.2 Commentary about Kiepert's algorithm . 49 7 Conclusion 51 7.1 Conclusion . 51 A Groups 53 B Permutation groups 54 C Rings 54 D Polynomials 55 References 57 2 CONTENTS 3 1 Introduction In this report, we study how one solves general polynomial equations using only the coef- ¯cients of the polynomial. Almost everyone who has taken some course in algebra knows that Galois proved that it is impossible to solve algebraic equations of degree ¯ve or higher, using only radicals. What is not so generally known is that it is possible to solve such equa- tions in another way, namely using techniques from the theory of elliptic functions. Galois theory and the theory of elliptic functions are both extensive and deep. But to understand and ¯nd a way of solving polynomial equations we do not need all this theory. The aim here is to pick out only the part of the theory which is necessary to understand why not all equations are solvable by radicals, and the alternative procedures by which we solve them. We will concentrate on ¯nding an algorithm for the quintic equation. In Chapter 2 we examine the history of polynomial equations. We also present the formulas for solutions to equations of degree less then ¯ve. A basic tool, the Tschirnhaus transformation, is introduced. These transformations are used to simplify the solution process by transforming a polynomial into another where some coe±cients are zero. In Chapter 3 we introduce some Galois theory. In Galois theory one examines the rela- tionship between the permutation group of the roots and a ¯eld containing the coe±cients of the polynomial. Here we present only as much theory as is necessary to understand why not all polynomial equations are solvable by radicals. The history of the quintic equation is also presented. In Chapter 4 we introduce the elliptic functions and examine their basic properties. We concentrate on one speci¯c elliptic function, namely the Weierstrass } function. We show that with the help of this function one can solve the quintic equation. However, this is just a theoretical possibility. To be able to get a practical algorithm, we need to know some facts about theta functions, with which we also deal in this chapter. The history of elliptic functions is included at the end of this chapter. Here we also present the general formula for solving any polynomial equation due to H. Umemura. This formula is also only of theoretical value, so we will look at other ways of solving the quintic. We present three such methods in the last chapter. The one that we will concentrate on is Kiepert's, because it uses both elliptic functions and Tschirnhaus transformations in a clear and straightforward way. In Chapter 5 we present two algorithms for solving the quintic, which are due to Hermite and Gordan. Hermite uses a more analytic method compared with Gordan's solution, which involves the use of the icosahedron. In Chapter 6 we present Kiepert's algorithm for the quintic. This lends itself to imple- mentation on a computer. This has in fact been done by B. King and R. Can¯eld, and at the end of the chapter we discuss their results. We conclude with an Appendix. Here one can ¯nd the most basic facts about groups, rings, and polynomials. Acknowledgment I would like to thank the following people: my supervisor Bengt Ove Turesson for helping me with this report and for teaching me LATEX 2" in which this report is written, Peter Hackman for helpfull hints about Galois theory, and Gunnar Fogelberg for historical background. I also thank Thomas Karlsson who told me that this examination work was available, and Bruce King for telling me about his work on the quintic. I also express my gratitude to John H. Conway, Keith Dennis, and William C. Waterhouse for giving me information about L. Kiepert. 4 1 INTRODUCTION 5 2 Quadratic, cubic, and quartic equations In this chapter we shall take a brief look at the history of equations of degree less than ¯ve. We will also study the techniques for solving these equations. One such technique, the Tschirnhaus transformation, will be examined in Section 2.3. 2.1 History Polynomial equations arise quite naturally in mathematics, when dealing with basic math- ematical problems, and so their study has a long history. The attempts to solve certain of these equations have lead to new and exciting theories in mathematics, with the result that polynomial equations has been an important cornerstone in the development to modern mathematics. It is known that in ancient Babylonia (2000 { 600 B.C.), one was able to solve the second order equation x2 + c x + c = 0; c R; 1 0 i 2 using the famous formula 2 c1 c1 x = c0: ¡ 2 § r 4 ¡ They could, of course, not deal with all of the roots, since they did not have access to the complex numbers. Archaeologists have found tablets with cuneiform signs, where problems concerning second order equations are dealt with. Starting with geometric problems, they were led to these equations by the Pythagorean theorem. The Babylonians also studied some special equations of third degree. The Greeks (600 B.C. { 300 A.D.) had a more geometric viewpoint of mathematics compared to the Babylonians, but they also considered second order equations. They made geometric solutions to the quadratic and constructed the segment x from known segments c0 and c1. Problems like trisecting an angle and doubling the volume of the cube led the Greeks to third degree equations which they tried to solve; but they had no general method to for solving such equations. The Hindus (600 B.C. { 1000 A.D.) invented the number zero, negative numbers, and the position system. Brahmagupta dealt with quadratic equations. The solutions were more algebraic than those of the Babylonians. The Arabs (500 B.C. { 1000 A.D.) enjoyed algebra. The famous al-Khwarizmi dealt with algebra in his book Al-jabr wa'l muqabala around 800 A.D. In this book, he exam- ined certain types of second order equations (and none of higher degree). Another Arab mathematician, Omar Khayyam (1048{1131) classi¯ed third degree equations and exam- ined their roots in some special cases. He also investigated equations of fourth degree. Fore more details about mathematics in these old cultures, see Boyer [2] and van der Waer- den [25]. Europe began to participate in the development of mathematics soon after the Arabs. The Italian Leonardo Fibonacci (1170{1250) wrote Liber Abacci, where he tried to solve quadratic equations in one or more variables. In Europe in the Middle Ages, competitions in mathematics were popular. This en- couraged the development of the art of solving equations. A new era began around the 6 2 QUADRATIC, CUBIC, AND QUARTIC EQUATIONS beginning of the ¯fteenth centuary in Italy, when Scipione del Ferro (1465{1526) succeeded in solving the equation 3 x + c1x + c0 = 0: In Section 2.3, we will show that every third degree equation can be transformed to the above form, which means that del Ferro had solved the general cubic equation. Before his death, del Ferro told his student Fior of his method for solving cubics. Fior challenged another Italian, Niccolo Tartaglia (1500{1557), to a mathematical duel. This forced Tartaglia, who knew that Fior could solve such equations, to ¯nd a general method for solving the cubic. Tartaglia succeded with this and thus won the competition. Tartaglia told Girolamo Cardano (1501{1576) about his method. Cardano published the method in his book Ars Magna (around 1540). Here he observed that, in certain cases, the method yields roots of negative numbers, and that these could be real numbers. This means that he was touching upon the complex numbers. Cardano made his student Lodovico Ferrari (1522{1565) solve the fourth degree equation, i.e., the quartic. Ferrari succeeded with this, and Cardano also published this method in Ars Magna. The case when Cardanos formulas lead to roots of negative numbers was studied by Rafael Bombelli (1526{1573) in his book Algebra in 1560. After the Italians, many famous mathematicians examined these equations, for example Francois Vi`ete (1540{1603) and Ehrenfried Walter von Tschirnhaus (1651{1708). It was ¯nally the German mathematician Gottfried von Leibniz (1646{1716) who, with- out any geometry, veri¯ed these formulas. More information about these Italians and the solutions to third and fourth degree equations can be found in Stillwell [24]. 2.2 Solving third and fourth degree equations Below we shall present the formulas for the solution of cubic and quartic equations.
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