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Elements of Abstract Algebra PDF

256 Pages·1984·4.37 MB·English
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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 Høegh-Krohn and Tom Lindstrøm. (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. (0-486-47172-1) PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, Bonnie Averbach and Orin Chein. (0-486-40917-1) THEORY OF LINEAR OPERATIONS, Stefan Banach. Translated by F. Jellett. (0-486-46983-2) VECTOR CALCULUS, Peter Baxandall and Hans Liebeck. (0-486-46620-5) INTRODUCTION TO VECTORS AND TENSORS: SECOND EDITION–TWO VOLUMES BOUND AS ONE, Ray M. Bowen and C.-C. Wang. (0-486-46914-X) ADVANCED TRIGONOMETRY, C. V. Durell and A. Robson. (0-486-43229-7) FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, Leon Ehrenpreis. (0-486-44975-0) THE THIRTEEN BOOKS OF THE ELEMENTS, Vol. 1, Euclid. Edited by Thomas L. Heath. (0-486- 60088-2) THE THIRTEEN BOOKS OF THE ELEMENTS, Vol. 2, Euclid. (0-486-60089-0) THE THIRTEEN BOOKS OF THE ELEMENTS, Vol. 3, Euclid. Edited by Thomas L. Heath. (0-486- 60090-4) AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, Stanley J. Farlow. (0-486-44595-X) PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, Stanley J. Farlow. (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 Grätzer. (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 Körner. (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

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This concise, readable, college-level text treats basic abstract algebra in remarkable depth and detail. An antidote to the usual surveys of structure, the book presents group theory, Galois theory, and classical ideal theory in a framework emphasizing proof of important theorems. Chapter I (Set The
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