Introductory Modern Algebra - A Historical Approach

Informationen zum Autor SAUL STAHL, PhD, is Professor in the Department of Mathematics at the University of Kansas. In addition to authoring six previous books and more than thirty papers in the field of geometry, Dr. Stahl has twice been the recipient of the Carl B. Allendoerfer Award from the Mathematical Association of America. Klappentext Praise for the First Edition"Stahl offers the solvability of equations from the historical point of view...one of the best books available to support a one-semester introduction to abstract algebra."- CHOICEIntroductory Modern Algebra: A Historical Approach, Second Edition presents the evolution of algebra and provides readers with the opportunity to view modern algebra as a consistent movement from concrete problems to abstract principles. With a few pertinent excerpts from the writings of some of the greatest mathematicians, the Second Edition uniquely facilitates the understanding of pivotal algebraic ideas.The author provides a clear, precise, and accessible introduction to modern algebra and also helps to develop a more immediate and well-grounded understanding of how equations lead to permutation groups and what those groups can inform us about such diverse items as multivariate functions and the 15-puzzle. Featuring new sections on topics such as group homomorphisms, the RSA algorithm, complex conjugation, the factorization of real polynomials, and the fundamental theorem of algebra, the Second Edition also includes:* An in-depth explanation of the principles and practices of modern algebra in terms of the historical development from the Renaissance solution of the cubic equation to Dedekind's ideals* Historical discussions integrated with the development of modern and abstract algebra in addition to many new explicit statements of theorems, definitions, and terminology* A new appendix on logic and proofs, sets, functions, and equivalence relations* Over 1,000 new examples and multi-level exercises at the end of each section and chapter as well as updated chapter summariesIntroductory Modern Algebra: A Historical Approach, Second Edition is an excellent textbook for upper-undergraduate courses in modern and abstract algebra. Zusammenfassung Praise for the First Edition "Stahl offers the solvability of equations from the historical point of view. one of the best books available to support a one-semester introduction to abstract algebra. Inhaltsverzeichnis Preface ix1 The Early History 11.1 The Breakthrough 12 Complex Numbers 92.1 Rational Functions of Complex Numbers 92.2 Complex Roots 172.3 Solvability by Radicals I 232.4 Ruler and Compass Constructibility 262.5 Orders of Roots of Unity 362.6 The Existence of Complex Numbers* 383 Solutions of Equations 453.1 The Cubic Formula 453.2 Solvability by Radicals II 493.3 Other Types of Solutions* 504 Modular Arithmetic 574.1 Modular Addition, Subtraction, and Multiplication 574.2 The Euclidean Algorithm and Modular Inverses 624.3 Radicals in Modular Arithmetic* 694.4 The Fundamental Theorem of Arithmetic* 705 The Binomial Theorem and Modular Powers 755.1 The Binomial Theorem 755.2 Fermat's Theorem and Modular Exponents 855.3 The Multinomial Theorem* 905.4 The Euler phi-Function* 926 Polynomials Over a Field 996.1 Fields and Their Polynomials 996.2 The Factorization of Polynomials 1076.3 The Euclidean Algorithm for Polynomials 1136.4 Elementary Symmetric Polynomials* 1196.5 Lagrange's Solution of the Quartic Equation* 1257 Galois Fields 1317.1 Galois's Construction of His Fields 1317.2 The Galois Polynomial 1397.3 The Primitive Element Theorem 1447.4 On the Variety of Galois Fields* 1478 Permutations 1558.1 Permuting the Variables of a Function I 1558.2 Permutations 1588.3 Permuting the Variables of a Function II 1668.4 The Parity of a Permutation 1699 Groups 1839.1 Permutation Groups 1839.2 Abstract Gr...

List of contents

Preface ix
 
1 The Early History 1
 
1.1 The Breakthrough 1
 
2 Complex Numbers 9
 
2.1 Rational Functions of Complex Numbers 9
 
2.2 Complex Roots 17
 
2.3 Solvability by Radicals I 23
 
2.4 Ruler and Compass Constructibility 26
 
2.5 Orders of Roots of Unity 36
 
2.6 The Existence of Complex Numbers* 38
 
3 Solutions of Equations 45
 
3.1 The Cubic Formula 45
 
3.2 Solvability by Radicals II 49
 
3.3 Other Types of Solutions* 50
 
4 Modular Arithmetic 57
 
4.1 Modular Addition, Subtraction, and Multiplication 57
 
4.2 The Euclidean Algorithm and Modular Inverses 62
 
4.3 Radicals in Modular Arithmetic* 69
 
4.4 The Fundamental Theorem of Arithmetic* 70
 
5 The Binomial Theorem and Modular Powers 75
 
5.1 The Binomial Theorem 75
 
5.2 Fermat's Theorem and Modular Exponents 85
 
5.3 The Multinomial Theorem* 90
 
5.4 The Euler phi-Function* 92
 
6 Polynomials Over a Field 99
 
6.1 Fields and Their Polynomials 99
 
6.2 The Factorization of Polynomials 107
 
6.3 The Euclidean Algorithm for Polynomials 113
 
6.4 Elementary Symmetric Polynomials* 119
 
6.5 Lagrange's Solution of the Quartic Equation* 125
 
7 Galois Fields 131
 
7.1 Galois's Construction of His Fields 131
 
7.2 The Galois Polynomial 139
 
7.3 The Primitive Element Theorem 144
 
7.4 On the Variety of Galois Fields* 147
 
8 Permutations 155
 
8.1 Permuting the Variables of a Function I 155
 
8.2 Permutations 158
 
8.3 Permuting the Variables of a Function II 166
 
8.4 The Parity of a Permutation 169
 
9 Groups 183
 
9.1 Permutation Groups 183
 
9.2 Abstract Groups 192
 
9.3 Isomorphisms of Groups and Orders of Elements 199
 
9.4 Subgroups and Their Orders 206
 
9.5 Cyclic Groups and Subgroups 215
 
9.6 Cayley's Theorem 218
 
10 Quotient Groups and their Uses 225
 
10.1 Quotient Groups 225
 
10.2 Group Homomorphisms 234
 
10.3 The Rigorous Construction of Fields 240
 
10.4 Galois Groups and Resolvability of Equations 253
 
11 Topics in Elementary Group Theory 261
 
11.1 The Direct Product of Groups 261
 
11.2 More Classifications 265
 
12 Number Theory 273
 
12.1 Pythagorean triples 273
 
12.2 Sums of two squares 278
 
12.3 Quadratic Reciprocity 285
 
12.4 The Gaussian Integers 293
 
12.5 Eulerian integers and others 304
 
12.6 What is the essence of primality? 310
 
13 The Arithmetic of Ideals 317
 
13.1 Preliminaries 317
 
13.2 Integers of a Quadratic Field 319
 
13.3 Ideals 322
 
13.4 Cancelation of Ideals 337
 
13.5 Norms of Ideals 341
 
13.6 Prime Ideals and Unique Factorization 343
 
13.7 Constructing Prime Ideals 347
 
14 Abstract Rings 355
 
14.1 Rings 355
 
14.2 Ideals 358
 
14.3 Domains 361
 
14.4 Quotients of Rings 367
 
A Excerpts: Al-Khwarizmi 377
 
B Excerpts: Cardano 383
 
C Excerpts: Abel 389
 
D Excerpts: Galois 395
 
E Excerpts: Cayley 401
 
F Mathematical Induction 405
 
G Logic, Predicates, Sets and Functions 413
 
G.1 Truth Tables 413
 
G.2 Modeling Implication 415
 
G.3 Predicates and their Negation 418
 
G.4 Two Applications 419
 
G.5 Sets 421
 
G.6 Functions 422
 
Biographies 427
 
Bib

Report

"This book is an excellent book for an upper-level, undergraduate, one or two semester course, in modern algebra, for a typical University student population that is not especially strong in proofs." ( MAA Reviews , 13 January 2014)

Product details

• Authors: Stahl, S Stahl, Saul Stahl, Stahl Saul
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 18.10.2013
• EAN: 9780470876169
• ISBN: 978-0-470-87616-9
• No. of pages: 464
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Mathematik, Algebra, Geschichte der Mathematik, Mathematics, History of mathematics, Moderne u. abstrakte Algebra, Modern/Abstract Algebra