Topics in algebra

Description

New edition includes extensive revisions of the material on finite groups and Galois Theory. New problems added throughout. Title Preface to the Second Edition Preface to the First Edition Contents 1. Preliminary Notions 1.1 Set Theory 1.2 Mappings 1.3 The Integers 2. Group Theory 2.1 Definition of a Group 2.2 Some Examples of Groups 2.3 Some Preliminary Lemmas 2.4 Subgroups 2.5 A Counting Principle 2.6 Normal Subgroups and Quotient Groups 2.7 Homomorphisms 2.8 Automorphisms 2.9 Cayley's Theorem 2.10 Permutation Groups 2.11 Another Counting Principle 2.12 Sylow's Theorem 2.13 Direct Products 2.14 Finite Abelian Groups 3. Ring Theory 3.1 Definition and Examples of Rings 3.2 Some Special Classes of Rings 3.3 Homomorphisms 3.4 Ideals and Quotient Rings 3.5 More Ideals and Quotient Rings 3.6 The Field of Quotients of an Integral Domain 3.7 Euclidean Rings 3.8 A Particular Euclidean Ring 3.9 Polynomial Rings 3.10 Polynomials over the Rational Field 3.11 Polynomial Rings over Commutative Rings 4. Vector Spaces and Modules 4.1 Elementary Basic Concepts 4.2 Linear Independence and Bases 4.3 Dual Spaces 4.4 Inner Product Spaces 4.5 Modules 5. Fields 5.1 Extension Fields 5.2 The Transcendence of e 5.3 Roots of Polynomials 5.4 Construction with Straightedge and Compass 5.5 More about Roots 5.6 The Elements of Galois Theory 5.7 Solvability by Radicals 5.8 Galois Groups over the Rationals 6. Linear Transformations 6.1 The Algebra of Linear Transformations 6.2 Characteristic Roots 6.3 Matrices 6.4 Canonical Forms: Triangular Form 6.5 Canonical Forms: Nilpotent Transformations 6.6 Canonical Forms: A Decomposition of V: Jordan Form 6.7 Canonical Forms: Rational Canonical Form 6.8 Trace and Transpose 6.9 Determinants 6.10 Hermitian, Unitary, and Normal Transformations 6.11 Real Quadratic Forms 7. Selected Topics 7.1 Finite Fields 7.2 Wedderburn's Theorem on Finite Division Rings 7.3 A Theorem of Frobenius 7.4 Integral Quaternions and the Four-Square Theorem Index