Algebra

Produktbeschreibung

Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material should be presented as general as possible, consistent with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises. 

Inhaltsverzeichnis

Introduction: Prerequisites and Preliminaries.- 1. Logic.- 2. Sets and Classes.- 3. Functions.- 4. Relations and Partitions.- 5. Products.- 6. The Integers.- 7. The Axiom of Choice, Order and Zorn's Lemma.- 8. Cardinal Numbers.- I: Groups.- 1. Semigroups, Monoids and Groups.- 2. Homomorphisms and Subgroups.- 3. Cyclic Groups.- 4. Cosets and Counting.- 5. Normality, Quotient Groups, and Homomorphisms.- 6. Symmetric, Alternating, and Dihedral Groups.- 7. Categories: Products, Coproducts, and Free Objects.- 8. Direct Products and Direct Sums.- 9. Free Groups, Free Products, Generators & Relations.- II: The Structure of Groups.- 1. Free Abelian Groups.- 2. Finitely Generated Abelian Groups.- 3. The Kruli-Schmidt Theorem.- 4. The Action of a Group on a Set.- 5. The Sylow Theorems.- 6. Classification of Finite Groups.- 7. Nilpotent and Solvable Groups.- 8. Normal and Subnormal Series.- III: Rings.- 1. Rings and Homomorphisms.- 2. Ideals.- 3. Factorization in Commutative Rings.- 4. Rings of Quotients and Localization.- 5. Rings of Polynomials and Formal Power Series.- 6. Factorization in Polynomial Rings.- IV: Modules.- 1. Modules, Homomorphisms and Exact Sequences.- 2. Free Modules and Vector Spaces.- 3. Projective and Injective Modules.- 4. Horn and Duality.- 5. Tensor Products.- 6. Modules over a Principal Ideal Domain.- 7. Algebras.- V: Fields and Galois Theory.- 1. Field Extensions.- 2. The Fundamental Theorem.- 3. Splitting Fields, Algebraic Closure and Normality.- 4. The Galois Group of a Polynomial.- 5. Finite Fields.- 6. Separability.- 7. Cyclic Extensions.- 8. Cyclotomic Extensions.- 9. Radical Extensions.- VI: The Structure of Fields.- 1. Transcendence Bases.- 2. Linear Disjointness and Separability.- VII: Linear Algebra.- 1. Matrices and Maps.- 2. Rank andEquivalence.- 3. Determinants.- 4. Decomposition of a Single Linear Transformation and Similarity.- 5. The Characteristic Polynomial, Eigenvectors and Eigenvalues.- VIII: Commutative Rings and Modules.- 1. Chain Conditions.- 2. Prime and Primary Ideals.- 3. Primary Decomposition.- 4. Noetherian Rings and Modules.- 5. Ring Extensions.- 6. Dedekind Domains.- 7. The Hilbert Nullstellensatz.- IX: The Structure of Rings.- 1. Simple and Primitive Rings.- 2. The Jacobson Radical.- 3. Semisimple Rings.- 4. The Prime Radical; Prime and Semiprime Rings.- 5. Algebras.- 6. Division Algebras.- X: Categories.- 1. Functors and Natural Transformations.- 2. Adjoint Functors.- 3. Morphisms.- List of Symbols. 

Kritik

From the book reviews:

"This is a text for a first-year graduate course in abstract algebra. It covers all the standard topics and has more than enough material for a year course." (Allen Stenger, MAA Reviews, September, 2014)

Thomas W. Hungerford

Algebra

"An excellent text from which to teach the beginning graduate survey course in algebra and I would recommend it to anyone considering a text for such a course."-LINEAR AND MULTILINEAR ALGEBRA