Introductory modern algebra : a historical approach / Saul Stahl, Department of Mathematics, University of Kansas.

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Bibliographic Details
Main Author: Stahl, Saul
Language:English
Published: Hoboken, New Jersey : John Wiley & Sons, Inc., [2013]
Edition:Second edition.
Subjects:
Algebra, Abstract.
Algebra.
Physical Description:xii, 447 pages ; 25 cm
Format: Book
Contents:
  • Chapter 1: The Early History; 1.1 The Breakthrough.
  • Chapter 2: Complex Numbers; 2.1 Rational Functions of Complex Numbers; 2.2 Complex Roots; 2.3 Solvability by Radicals I; 2.4 Ruler-and-Compass Constructibility of Regular Polygons; 2.5 Orders of Roots of Unity; 2.6 The Existence of Complex Numbers.
  • Chapter 3: Solutions of Equations; 3.1 The Cubic Formula; 3.2 Solvability by Radicals II; 3.3 Other Types of Solutions.
  • Chapter 4: Modular Arithmetic; 4.1 Modular Addition, Subtraction, and Multiplication; 4.2 The Euclidean Algorithm and Modular Inverses; 4.3 Radicals in Modular Arithmetic; 4.4 The Fundamental Theorem of Arithmetic.
  • Chapter 5: The Binomial Theorem and Modular Powers; 5.1 The Binomial Theorem; 5.2 Fermat's Theorem and Modular Exponents; 5.3 The Multinomial Theorem; 5.4 The Euler ₁їFunction.
  • Chapter 6: Polynomials Over A Field; 6.1 Fields and Their Polynomials; 6.2 The Factorization of Polynomials; 6.3 The Euclidean Algorithm for Polynomials; 6.4 Elementary Symmetric Polynomials; 6.5 Lagrange's Solution of the Quartic Equation.
  • Chapter 7: Galois Fields; 7.1 Galois's Construction of His Fields7.2 The Galois Polynomial; 7.3 The Primitive Element Theorem; 7.4 On the Variety of Galois Fields.
  • Chapter 8: Permutations; 8.1 Permuting the Variables of a Function I; 8.2 Permutations; 8.3 Permuting the Variables of a Function II; 8.4 The Parity of a Permutation.
  • Chapter 9: Groups; 9.1 Permutation Groups; 9.2 Abstract Groups; 9.3 Isomorphisms of Groups and Orders of Elements; 9.4 Subgroups and Their Orders; 9.5 Cyclic Groups and Subgroups; 9.6 Cayley's Theorem.
  • Chapter 10: Quotient Groups and Their Uses; 10.1 Quotient Groups; 10.2 Group Homomorphisms; 10.3 The Rigorous Construction of Fields10.4 Galois Groups and the Resolvability of Equations.
  • Chapter 11: Topics in Elementary Group Theory; 11.1 The Direct Product of Groups; 11.2 More Classifications.
  • Chapter 12: Number Theory; 12.1 Pythagorean Triples; 12.2 Sums of Two Squares; 12.3 Quadratic Reciprocity; 12.4 The Gaussian Integers; 12.5 Eulerian Integers and Others; 12.6 What Is the Essence of Primality?
  • Chapter 13: The Arithmetic of Ideals; 13.1 Preliminaries; 13.2 Integers of a Quadratic Field; 13.3 Ideals; 13.4 Cancelation of Ideals; 13.5 Norms of Ideals; 13.6 Prime Ideals and Unique Factorization13.7 Constructing Prime Ideals.
  • Chapter 14: Abstract Rings; 14.1 Rings; 14.2 Ideals; 14.3 Domains; 14.4 Quotients of Rings.
  • A. Excerpts from Al-Khwarizmi's Solution of the Quadratic Equation1; B. Excerpts from Cardano's Ars Magna1; C. Excerpts from Abel's A Demonstration of the Impossibility of the Algebraic Resolution of General Equations Whose Degree Exceeds Four1; D. Excerpts from Galois's On the Theory of Numbers1; E. Excerpts from Cayley's The Theory of Groups1; F. Mathematical Induction; G. Logic, Predicates, Sets, and Functions.