Introductory Modern Algebra A Historical Approach 2nd Edition by Saul Stahl 0470876166 9780470876169

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Product details:

ISBN 10: 0470876166 

ISBN 13: 9780470876169

Author: Saul Stahl

This book presents pivotal ideas in the evolution of algebra and provides readers with the opportunity to view the evolution of modern algebra as a consistent movement from concrete problems to abstract principles. Praise for the First Edition: “In this excellent presentation of modern algebra, 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.”

Table of contents:

Chapter 1 THE EARLY HISTORY
1.1 The Breakthrough
Proposition 1.3
Exercises 1.1
Chapter Summary
Chapter Review Exercises
New Terms

Chapter 2 COMPLEX NUMBERS
2.1 Rational Functions of Complex Numbers
Figure 2.I Complex addition
Proposition 2.1
Figure 2.2 The argument and the modulus
Figure 2.3 Some complex numbers
Theorem 2.2
Corollary 2.3
Theorem 2.4
Exercises 2.1
2.2 Complex Roots
Figure 2.4 Complex roots of unity
Theorem 2.5
Proposition 2.6
Proposition 2.7
Proposition 2.8
Exercises 2.2
2.3 Solvability by Radicals I
Exercises 2.3
2.4 Ruler-and-Compass Constructibility of Regular Polygons
Figure 2.5 A square and a regular octagon
Figure 2.6 An equilateral triangle and a regular pentagon
Proposition 2.10
Proposition 2.11
Lemma 2.12
Figure 2.7 The multiplication of constructible numbers
Figure 2.8 The reciprocal of a constructible number
Figure 2.9 The square root of a constructible number
Corollary 2.13
Proposition 2.14
Figure 2.10 The regular 17-gon
Theorem 2.15
Exercises 2.4
2.5 Orders of Roots of Unity
Proposition 2.16
Corollary 2.17
Exercises 2.5
2.6 The Existence of Complex Numbers
Exercises 2.6
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 3 SOLUTIONS OF EQUATIONS
3.1 The Cubic Formula
Theorem 3.1
Exercises 3.1
3.2 Solvability by Radicals II
3.3 Other Types of Solutions
Theorem 3.9
Theorem 3.10
Figure 3.I The Newton-Raphson method
Exercises 3.3
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 4 MODULAR ARITHMETIC
4.1 Modular Addition, Subtraction, and Multiplication
Table 4.I Arithmetic modulo 4
Table 4.2 Arithmetic modulo 5
Table 4.3 Arithmetic modulo 6
Table 4.4 Arithmetic modulo 7
Exercises 4.1
4.2 The Euclidean Algorithm and Modular Inverses
Proposition 4.1
Corollary 4.4
Proposition 4.5
Lemma 4.6
Corollary 4.7
Theorem 4.8
Exercises 4.2
4.3 Radicals in Modular Arithmetic
Exercises 4.3
4.4 The Fundamental Theorem of Arithmetic
Theorem 4.9
Exercises 4.4
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 5 THE BINOMIAL THEOREM AND MODULAR POWERS
5.1 The Binomial Theorem
Theorem 5.3
Figure 5.I Pascal’s Triangle
Figure 5.2 Pascal’s Triangle as depicted in a Chinese book in 1303
Proposition 5.10
Exercises 5.1
Figure 5.3 The grid G3,2
Figure 5.4 Triangle counting problems
5.2 Fermat’s Theorem and Modular Exponents
Proposition 5.14
Theorem 5.15
Corollary 5.16
Table 5.I The elements of ℤ7 and their orders
Proposition 5.17
Corollary 5.18
Proposition 5.19
Proposition 5.20
Exercises 5.2
5.3 The Multinomial Theorem
Theorem 5.21
Exercises 5.3
5.4 The Euler ϕ-Function
Lemma 5.23
Theorem 5.25
Table 5.2 rsa encryption
Exercises 5.4
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 6 POLYNOMIALS OVER A FIELD
6.1 Fields and Their Polynomials
Proposition 6.1
Theorem 6.2
Proposition 6.3
Proposition 6.4
Exercises 6.1
6.2 The Factorization of Polynomials
Proposition 6.6
Corollary 6.7
Proposition 6.8
Proposition 6.9
Lemma 6.10
Corollary 6.11
Exercises 6.2
6.3 The Euclidean Algorithm for Polynomials
Lemma 6.12
Figure 6.I The Euclidean algorithm for polynomials
Proposition 6.14
Proposition 6.15
Exercises 6.3
6.4 Elementary Symmetric Polynomials
Theorem 6.19
Proposition 6.20
Corollary 6.21
Exercises 6.4
6.5 Lagrange’s Solution of the Quartic Equation
Theorem 6.22
Exercises 6.5
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 7 GALOIS FIELDS
7.1 Axioms for a Field
7.2 Axioms for a Finite Field
Table 7.I Addition and multiplication in ℤ3
Theorem 7.1
Corollary 7.2
Theorem 7.3
Theorem 7.4
Corollary 7.5
7.3 Structure of a Galois Field
Theorem 7.6
Theorem 7.7
Exercises 7.1
7.4 Polynomial Representation of a Galois Field
Theorem 7.8
Exercises 7.2
7.5 Construction of a Galois Field
Theorem 7.9
Corollary 7.10
Corollary 7.11
Proposition 7.12
Proposition 7.13
Exercises 7.3
7.6 Properties of Galois Fields
Theorem 7.14
Exercises 7.4
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 8 PERMUTATIONS AND GROUPS
8.1 Permutations
Theorem 8.1
Exercises 8.1
8.2 Groups
Table 8.I A group table
Exercises 8.2
8.3 Subgroups and Generators
Theorem 8.2
Exercises 8.3
8.4 Properties of Groups
Theorem 8.3
Theorem 8.4
Corollary 8.5
Theorem 8.6
Corollary 8.7
Corollary 8.8
Exercises 8.4
8.5 Cosets and Quotient Groups
Theorem 8.9
Corollary 8.10
Theorem 8.11
Corollary 8.12
Exercises 8.5
8.6 Permutation Groups
Proposition 8.13
Corollary 8.14
Exercises 8.6
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 9 POLYNOMIAL EQUATIONS AND SYMMETRY
9.1 Symmetries of the Roots
Theorem 9.1
Proposition 9.2
Exercises 9.1
9.2 The Galois Group
Theorem 9.3
Corollary 9.4
Proposition 9.5
Exercises 9.2
9.3 Galois Groups and Field Extensions
Theorem 9.6
Theorem 9.7
Corollary 9.8
Exercises 9.3
9.4 Fundamental Theorem of Galois Theory
Theorem 9.9
Exercises 9.4
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 10 THE ALGEBRA OF MATRICES
10.1 Matrices
10.2 Matrix Arithmetic
Proposition 10.1
10.3 Determinants
10.4 Inverses
Proposition 10.2
10.5 Special Types of Matrices
Theorem 10.3
Exercises
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 11 VECTOR SPACES
11.1 Axioms for a Vector Space
Proposition 11.1
11.2 Subspaces
Proposition 11.2
11.3 Linear Independence, Basis, and Dimension
Theorem 11.3
Theorem 11.4
Corollary 11.5
Exercises
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 12 LINEAR TRANSFORMATIONS AND MATRICES
12.1 Linear Transformations
Proposition 12.1
Theorem 12.2
12.2 Matrices and Linear Transformations
Proposition 12.3
Theorem 12.4
Corollary 12.5
12.3 Similar Matrices
Theorem 12.6
12.4 The Characteristic Equation
Theorem 12.7
12.5 Eigenvalues and Eigenvectors
Theorem 12.8
Exercises
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

Chapter 13 CANONICAL FORMS
13.1 Diagonalization
Theorem 13.1
13.2 The Cayley-Hamilton Theorem
Theorem 13.2
13.3 Jordan Form
Theorem 13.3
13.4 Rational Canonical Form
13.5 Applications
Exercises
Chapter Summary
Chapter Review Exercises
New Terms
Supplementary Exercises

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Tags: Saul Stahl, Introductory, Modern, Historical