Introductory Modern Algebra

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."

--CHOICE

Introductory 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 summaries

Introductory Modern Algebra: A Historical Approach, Second
Edition is an excellent textbook for upper-undergraduate
courses in modern and abstract algebra.

Auteur

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.

Texte du rabat

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."— CHOICE

Introductory 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 summaries
  Introductory Modern Algebra: A Historical Approach, Second Edition is an excellent textbook for upper-undergraduate courses in modern and abstract algebra.

Résumé
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."
CHOICE

Introductory 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 summaries
  Introductory Modern Algebra: A Historical Approach, Second Edition is an excellent textbook for upper-undergraduate courses in modern and abstract algebra.

Contenu

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 -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 …