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Helping to make the study of modern algebra more accessible, this text gradually introduces and develops concepts through helpful features that provide guidance on the techniques of proof construction and logic analysis. The text develops mathematical maturity for students by presenting the material in a theorem-proof format, with definitions and major results easily located through a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and of providing a bridge to higher-level mathematical courses.
1. FUNDAMENTALS. Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations. Key Words and Phrases. A Pioneer in Mathematics: Arthur Cayley. 2. THE INTEGERS. Postulates for the Integers (Optional). Mathematical Induction. Divisibility. Prime Factors and Greatest Common Divisor. Congruence of Integers. Congruence Classes. Introduction to Coding Theory (Optional). Introduction to Cryptography (Optional). Key Words and Phrases. A Pioneer in Mathematics: Blaise Pascal. 3. GROUPS. Definition of a Group. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms. Key Words and Phrases. A Pioneer in Mathematics: Niels Henrik Abel. 4. MORE ON GROUPS. Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional). Key Words and Phrases. A Pioneer in Mathematics: Augustin Louis Cauchy. 5. RINGS, INTEGRAL DOMAINS, AND FIELDS. Definition of a Ring. Integral Domains and Fields. The Field of Quotients of an Integral Domain. Ordered Integral Domains. Key Words and Phrases. A Pioneer in Mathematics: Richard Dedekind. 6. MORE ON RINGS. Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring. Maximal Ideals (Optional). Key Words and Phrases. A Pioneer in Mathematics: Amalie Emmy Noether. 7. REAL AND COMPLEX NUMBERS. The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers. Key Words and Phrases. A Pioneer in Mathematics: William Rowan Hamilton. 8. POLYNOMIALS. Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in F[x]. Zeros of a Polynomial. Solutions of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field. Key Words and Phrases. A Pioneer in Mathematics: Carl Friedrich Gauss. Appendix: The Basics Of Logic. Bibliography. Answers to Selected Computational Exercises. Index.