# 新河南快赢481开奖视: ElementsofAbstractAlgebra-AllanClark

DOVER BOOKS ON MATHEMATICS HANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES, Edited by Milton Abramowitz and Irene A. Stegun. (0-486-61272-4) ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, Jiri Adamek, Horst Herrlich, George E. Strecker. (0-486-46934-4) NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, Sergio Albeverio, Jens Erik Fenstad, Raphael H?egh-Krohn and Tom Lindstr?m. (0-486-46899-2) MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent’ev. (0-486-40916-3) COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THE CIRCLE, Nathan Altshiller-Court. (0-486-45805-9) THE WORKS OF ARCHIMEDES, Archimedes. Translated by Sir Thomas Heath. (0-486-42084-1) REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, Robert B. Ash. (0-486-47220-5) INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, Louis Auslander and Robert E. MacKenzie. 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(0-486-67620-X) STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, Avner Friedman. (0-486-45359-6) ADVANCED CALCULUS, Avner Friedman. (0-486-45795-8) POINT SET TOPOLOGY, Steven A. Gaal. (0-486-47222-1) DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. Gardiner. (0-486-45299-9) LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, George Gr?tzer. (0-486-47173-X) ORDINARY DIFFERENTIAL EQUATIONS, Jack K. Hale. (0-486-47211- 6) METHODS OF APPLIED MATHEMATICS, Francis B. Hildebrand. (0-486- 67002-3) BASIC ALGEBRA I: SECOND EDITION, Nathan Jacobson. (0-486-47189- 6) BASIC ALGEBRA II: SECOND EDITION, Nathan Jacobson. (0-486- 47187-X) NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY THE FINITE ELEMENT METHOD, Claes Johnson. (0-486-46900-X) ADVANCED EUCLIDEAN GEOMETRY, Roger A. Johnson. (0-486- 46237-4) GEOMETRY AND CONVEXITY: A STUDY IN MATHEMATICAL METHODS, Paul J. Kelly and Max L. Weiss. (0-486-46980-8) TRIGONOMETRY REFRESHER, A. Albert Klaf. (0-486-44227-6) CALCULUS: AN INTUITIVE AND PHYSICAL APPROACH (SECOND EDITION), Morris Kline. (0-486-40453-6) THE PHILOSOPHY OF MATHEMATICS: AN INTRODUCTORY ESSAY, Stephan K?rner. (0-486-47185-3) COMPANION TO CONCRETE MATHEMATICS: MATHEMATICAL TECHNIQUES AND VARIOUS APPLICATIONS, Z. A. Melzak. (0-486- 45781-8) NUMBER SYSTEMS AND THE FOUNDATIONS OF ANALYSIS, Elliott Mendelson. (0-486-45792-3) EXPERIMENTAL STATISTICS, Mary Gibbons Natrella. (0-486-43937-2) AN INTRODUCTION TO IDENTIFICATION, J. P. Norton. (0-486-46935- 2) BEYOND GEOMETRY: CLASSIC PAPERS FROM RIEMANN TO EINSTEIN, Edited with an Introduction and Notes by Peter Pesic. (0-486- 45350-2) THE STANFORD MATHEMATICS PROBLEM BOOK: WITH HINTS AND SOLUTIONS, G. Polya and J. Kilpatrick. (0-486-46924-7) SPLINES AND VARIATIONAL METHODS, P. M. Prenter. (0-486-46902- 6) PROBABILITY THEORY, A. Renyi. (0-486-45867-9) LOGIC FOR MATHEMATICIANS, J. Barkley Rosser. (0-486-46898-4) PARTIAL DIFFERENTIAL EQUATIONS: SOURCES AND SOLUTIONS, Arthur David Snider. (0-486-45340-5) INTRODUCTION TO BIOSTATISTICS: SECOND EDITION, Robert R. Sokal and F. James Rohlf. (0-486-46961-1) MATHEMATICAL PROGRAMMING, Steven Vajda. (0-486-47213-2) THE LOGIC OF CHANCE, John Venn. (0-486-45055-4) THE CONCEPT OF A RIEMANN SURFACE, Hermann Weyl. (0-486- 47004-0) INTRODUCTION TO PROJECTIVE GEOMETRY, C. R. Wylie, Jr. (0-486- 46895-X) FOUNDATIONS OF GEOMETRY, C. R. Wylie, Jr. (0-486-47214-0) See every Dover book in print at www.doverpublications.com For my parents I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the more general character of the expression a + b = c. We are mere operatives, empirics, and egotists until we learn to think in letters instead of figures. OLIVER WENDELL HOLMES The Autocrat of the Breakfast Table Copyright ? 1971, 1984 by Allan Clark. All rights reserved. This Dover edition, first published in 1984, is an unabridged and corrected republication of the work first published by Wadsworth Publishing. Company, Belmont, California, in 1971. Library of Congress Cataloging in Publication Data Clark, Allan, 1935– Elements of abstract algebra. “Corrected republication”—Verso t.p. Originally published: Belmont, Calif. : Wadsworth, ? 1971. Bibliography: p. Includes index. 1. Algebra, Abstract. 1. Title. [QA162.C57 1984] 512′.02 84-6118 9780486140353 Manufactured in the United States by Courier Corporation 64725011 www.doverpublications.com Foreword Modern or “abstract” algebra is widely recognized as an essential element of mathematical education. Moreover, it is generally agreed that the axiomatic method provides the most elegant and efficient technique for its study. One must continually bear in mind, however, that the axiomatic method is an organizing principle and not the substance of the subject. A survey of algebraic structures is liable to promote the misconception that mathematics is the study of axiom systems of arbitrary design. It seems to me far more interesting and profitable in an introductory study of modern algebra to carry a few topics to a significant depth. Furthermore I believe that the selection of topics should be firmly based on the historical development of the subject. This book deals with only three areas of abstract algebra: group theory, Galois theory, and classical ideal theory. In each case there is more depth and detail than is customary for a work of this type. Groups were the first algebraic structure characterized axiomatically. Furthermore the theory of groups is connected historically and mathematically to the Galois theory of equations, which is one of the roots of modern algebra. Galois theory itself gives complete answers to classical problems of geometric constructibility and solvability of equations in radicals. Classical ideal theory, which arose from the problems of unique factorization posed by Fermat’s last theorem, is a natural sequel to Galois theory and gives substance to the study of rings. All three topics converge in the fundamental theorem of algebraic number theory for Galois extensions of the rational field, the final result of the book. Emil Artin wrote: We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment’s glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits visualization of the whole, and yet this logical structure must predominate or chaos would result. 1 A text must cling stubbornly to the logical sequence of the subject. A lecturer may be peripatetic, frequently with engaging results, but an author must tread a straight and narrow path. However, though written sequentially, this book need not be read that way. The material is broken into short articles, numbered consecutively throughout. These can be omitted, modified, postponed until needed, or given for outside reading. Most articles have exercises, a very few of which are used later in proofs. What can be covered in an ordinary course and for what students the text is suitable are questions left to the instructor, who is the best judge of local conditions. It is helpful, but certainly not essential, for the reader to know a little linear algebra for the later chapters—in particular Cramer’s rule. (Vector spaces, bases, and dimension are presented in articles 90–95.) Finally, I must gratefully acknowledge the assistance of Mrs. Theodore Weller and Miss Elizabeth Reynolds, who typed the manuscript, and the help of Messrs. George Blundall and John Ewing, who gave their time and patience to proofing it. Providence, Rhode Island January 1, 1970 Table of Contents DOVER BOOKS ON MATHEMATICS Title Page Dedication Copyright Page Foreword Introduction Chapter 1 - Set Theory Chapter 2 - Group Theory Chapter 3 - Field Theory Chapter 4 - Galois Theory Chapter 5 - Ring Theory Chapter 6 - Classical Ideal Theory Bibliography Index Introduction Classical algebra was the art of resolving equations. Modern algebra, the subject of this book, appears to be a different science entirely, hardly concerned with equations at all. Yet the study of abstract structure which characterizes modern algebra developed quite naturally out of the systematic investigation of equations of higher degree. What is more, the modern abstraction is needed to bring the classical theory of equations to a final perfect form. The main part of this text presents the elements of abstract algebra in a concise, systematic, and deductive framework. Here we shall trace in a leisurely, historical, and heuristic fashion the genesis of modern algebra from its classical origins. The word algebra comes from an Arabic word meaning “reduction” or “restoration.” It first appeared in the title of a book by Muhammad ibn Musa al-Khwarizmi about the year 825 A.D. The renown of this work, which gave complete rules for solving quadratic equations, led to use of the word algebra for the whole science of equations. Even the author’s name lives on in the word algorithm (a rule for reckoning) derived from it. Up to this point the theory of equations had been a collection of isolated cases and special methods. The work of al-Khwarizmi was the first attempt to give it form and unity. The next major advance came in 1545 with the publication of Artis Magnae sive de Regulis Algebraicis by Hieronymo Cardano (1501–1576). Cardano’s book, usually called the Ars Magna, or “The Grand Art,” gave the complete solution of equations of the third and fourth degree. Exactly how much credit for these discoveries is due to Cardano himself we cannot be certain. The solution of the quartic is due to Ludovico Ferrari (1522–1565), Cardano’s student, and the solution of the cubic was based in part upon earlier work of Scipione del Ferro (1465?–1526). The claim of Niccolo Fontana (1500?–1557), better known as Tartaglia (“the stammerer”), that he gave Cardano the cubic under a pledge of secrecy, further complicates the issue. The bitter feud between Cardano and Tartaglia obscured the true primacy of del Ferro. A solution of the cubic equation leading to Cardano’s formula is quite simple to give and motivates what follows. The method we shall use is due to Hudde, about 1650. Before we start, however, it is necessary to recall that every complex number has precisely three cube roots. For example, the complex number 1 = 1 + 0i has the three cube roots, l (itself), , and . In general, if z is any one of the cube roots of a complex number w, then the other two are ωz and ω2z. For simplicity we shall consider only a special form of the cubic equation, (1) (However, the general cubic equation may always be reduced to one of this form without difficulty.) First we substitute u + v for x to obtain a new equation, (2) which we rewrite as (3) Since we have substituted two variables, u and v, in place of the one variable x, we are free to require that 3uv + q = 0, or in other words, that v = ?q/3u. We use this to eliminate v from (3), and after simplification we obtain, (4) This last equation is called the resolvent equation of the cubic (1). We may view it as a quadratic equation in u3 and solve it by the usual method to obtain (5) Of course a complete solution of the two equations embodied in (5) gives six values of u—three cube roots for each choice of sign. These six values of u are the roots of the sixth-degree resolvent (4). We observe however that if u is a cube root of , then v = ?q/3u is a cube root of Consequently the six roots of (4) may be conveniently designated as u, ωu, ω2u and v, ωv, ω2v, where uv = ?q/3. Thus the three roots of the original equation are (6) where In other words, the roots of the original cubic equation (1) are given by the formula of Cardano, in which the cube roots are varied so that their product is always ? q/3. For our purposes we do not need to understand fully this complete solution of the cubic equation—only the general pattern is of interest here. The important fact is that the roots of the cubic equation can be expressed in terms of the roots of a resolvent equation which we know how to solve. The same fact is true of the general equation of the fourth degree. For a long time mathematicians tried to find a solution of the general quintic, or fifth-degree, equation without success. No method was found to carry them beyond the writings of Cardano on the cubic and quartic. Consequently they turned their attention to other aspects of the theory of equations, proving theorems about the distribution of roots and finding methods of approximating roots. In short, the theory of equations became analytic. One result of this approach was the discovery of the fundamental theorem of algebra by D’Alembert in 1746. The fundamental theorem states that every algebraic equation of degree n has n roots. It implies, for example, that the equation xn ? 1 = 0 has n roots—the so-called nth roots of unity—from which it follows that every complex number has precisely n nth roots. D’Alembert’s proof of the fundamental theorem was incorrect (Gauss gave the first correct proof in 1799) but this was not recognized for many years, during which the theorem was popularly known as “D’Alembert’s theorem.” D’Alembert’s discovery made it clear that the question confronting algebraists was not the existence of solutions of the general quintic equation, but whether or not the roots of such an equation could be expressed in terms of its coefficients by means of formulas like those of Cardano, involving only the extraction of roots and the rational operations of addition, subtraction, multiplication, and division. In a new attempt to resolve this question Joseph Louis Lagrange (1736– 1813) undertook a complete restudy of all the known methods of solving cubic and quartic equations, the results of which he published in 1770 under the title Réflexions sur la résolution algébrique des equations. Lagrange observed that the roots of the resolvent equation of the cubic (4) can be expressed in terms of the roots α1, α2, α3 of the original equation (1) in a completely symmetric fashion. Specifically, (7) All these expressions may be obtained from any one of them by permuting the occurrences of α1, α2, α3 in all six possible ways. Lagrange’s observation was important for several reasons. We obtained the resolvent of the cubic by making the substitution x = u + v. Although this works quite nicely, there is no particular rhyme nor reason to it—it is definitely ad hoc. However Lagrange’s observation shows how we might have constructed the resolvent on general principles and suggests a method for constructing resolvents of equations of higher degrees. Furthermore it shows that the original equation is solvable in radicals if and only if the resolvent equation is. To be explicit let us consider a quartic equation, (8) and suppose that the roots are the unknown complex numbers α1 α2, α3, α4. Without giving all the details we shall indicate how to construct the resolvent equation. First we recall that the fourth roots of unity are the complex numbers 1, i, i2, i3, where and i2 = ? 1, i3 = ?i. Then the roots of the resolvent are the twenty-four complex numbers (9) where the indices i, j, k, l are the numbers 1, 2, 3, 4 arranged in some order. Therefore the resolvent equation is the product of the twenty-four distinct factors (x ? uijkl). That is, we may write the resolvent equation in the form (10) Thus the resolvent of the quartic has degree 24, and it would seem hopeless to solve. It turns out, however, that every exponent of x in φ(x) is divisible by 4, and consequently φ(x) = 0 may be viewed as a sixth-degree equation in x4. What is more, this sixth-degree equation can be reduced to the product of two cubic equations (in a way we cannot make explicit here). Since cubics can be solved, a solution of the quartic can be obtained by a specific formula in radicals. (Such a formula is so unwieldy that it is more useful and understandable simply to describe the process for obtaining solutions.) For quintic, or fifth-degree, equations Lagrange’s theory yields a resolvent equation of degree 120, which is a 24th-degree equation in x5. Lagrange was convinced that his approach, which revealed the similarities in the resolution of cubics and quartics, represented the true metaphysics of the theory of equations. The difficulty of the computations prevented Lagrange from testing whether his techniques could produce a formula for resolving the quintic in radicals. Moreover, with his new insights, Lagrange could foresee the point at which the process might break down, and he gave equal weight to the impossibility of such a formula. A short time afterward, Paolo Ruffini (1765–1822) published a proof of the unsolvability of quintic equations in radicals. Ruffini’s argument, given in his two-volume Teoria generale delle equazioni of 1799, was correct in essence, but was not, in actual fact, a proof. A complete and correct proof was given by Niels Henrik Abel (1802–1829) in 1826 in a small book published at his own expense. The brilliant work of Abel closed the door on a problem which had excited and frustrated the best mathematical minds for almost three centuries. There remained one final step. Some equations of higher degree are clearly solvable in radicals even though they cannot be factored. Abel’s theorem raised the question: which equations are solvable in radicals and which are not? The genius évariste Galois (1811–1832) gave a complete answer to this question in 1832. Galois associated to each algebraic equation a system of permutations of its roots, which he called a group. He was able to show equivalence of the solvability of an equation in radicals, with a property of its group. Thus he made important discoveries in the theory of groups as well as the theory of equations. Unfortunately Galois’ brief and tragic life ended in a foolish duel before his work was understood. His theory perfected the ideas of Lagrange, Ruffini, and Abel and remains one of the stunning achievements of modern mathematical thought. At this point we can only leave as a mystery the beautiful relation Galois discovered between the theory of equations and the theory of groups—a mystery resolved by the deep study of both theories undertaken in the text. We can, however, gain some insight into modern abstraction by a short and informal discussion of groups. To take