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Abstract Algebra (2nd Edition) by Pierre Antoine Grillet



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Abstract Algebra (2nd Edition) written by Pierre Antoine Grillet, Dept. Mathematics, Tulane University, USA, grillet@math.tulane.edu. About the first edition:"The text is geared to the needs of the beginning graduate student, covering with complete, well-written proofs the usual major branches of groups, rings, fields, and modules...[n]one of the material one expects in a book like this is missing, and the level of detail is appropriate for its intended audience. (Alberto Delgado, MathSciNet)"This text promotes the conceptual understanding of algebra as a whole, and that with great methodological mastery. Although the presentation is predominantly abstract...it nevertheless features a careful selection of important examples, together with a remarkably detailed and strategically skillful elaboration of the more sophisticated, abstract theories. (Werner Kleinert, Zentralblatt)For the new edition, the author has completely rewritten the text, reorganized many of the sections, and even cut or shortened material which is no longer essential. He has added a chapter on Ext and Tor, as well as a bit of topology. "Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra. The first chapters blend standard contents with a careful introduction to proofs with arrows. The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra. There are over 1400 exercises, at varying degrees of difficulty.

Abstract Algebra (2nd Edition) written by Pierre Antoine Grillet cover the following topics.

  • I.Groups
    1. Semigroups
    2. Groups
    3. Subgroups
    4. Homomorphisms
    5. The Isomorphism Theorems
    6. Free Groups
    *8. Free Products

  • II. Structure of Groups
    1. Direct Products
    *2. The Krull-Schmidt Theorem
    3. Group Actions
    4. Symmetric Groups
    5. The Sylow Theorems
    6. Small Groups
    7. Composition Series
    *8. The General Linear Group
    9. Solvable Groups
    *10. Nilpotent Groups
    *11. Semidirect Products
    *12. Group Extensions

  • III.Rings
    1. Rings
    2. Subrings and Ideals
    3. Homomorphisms
    4. Domains and Fields
    5. Polynomials in One Variable
    6. Polynomials in Several Variables
    *7. Formal Power Series
    8. Principal Ideal Domains
    *9. Rational Fractions
    10. Unique Factorization Domains
    11. Noetherian Rings
    12. Gr¨obner Bases

  • IV.Field Extensions
    1. Fields
    2. Extensions
    3. Algebraic Extensions
    4. The Algebraic Closure
    5. Separable Extensions
    6. Purely Inseparable Extensions
    7. Resultants and Discriminants
    8. Transcendental Extensions
    9. Separability

  • V.Galois Theory
    1. Splitting Fields
    2. Normal Extensions
    3. Galois Extensions
    4. Infinite Galois Extensions
    5. Polynomials
    6. Cyclotomy
    7. Norm and Trace
    8. Solvability by Radicals
    9. Geometric Constructions

  • VI. Fields with Orders or Valuations
    1. Ordered Fields
    2. Real Fields
    3. Absolute Values
    4. Completions
    5. Extensions
    6. Valuations
    7. Extending Valuations
    8. Hensel’s Lemma
    9. Filtrations and Completions

  • VII.Commutative Rings
    1. Primary Decomposition
    2. Ring Extensions
    3. Integral Extensions
    4. Localization
    5. Dedekind Domains
    6. Algebraic Integers
    7. Galois Groups
    8. Minimal Prime Ideals
    9. Krull Dimension
    10. Algebraic Sets
    11. Regular Mappings

  • VIII.Modules
    1. Definition
    2. Homomorphisms
    3. Direct Sums and Products
    4. Free Modules
    5. Vector Spaces
    6. Modules over Principal Ideal Domains
    7. Jordan Form ofMatrices
    8. Chain Conditions
    9. Gr¨obner Bases

  • IX.Semisimple Rings And Modules
    1. Simple Rings and Modules
    2. Semisimple Modules
    3. The Artin-Wedderburn Theorem
    4. Primitive Rings
    5. The Jacobson Radical
    6. Artinian Rings
    7. Representations of Groups
    8. Characters
    9. Complex Characters

  • X. Projectives and Injectives
    1. Exact Sequences
    2. Pullbacks and Pushouts
    3. Projective Modules
    4. Injective Modules
    5. The Injective Hull
    6. Hereditary Rings

  • XI.Constructions
    1. Groups of Homomorphisms
    2. Properties of Hom
    3. Direct Limits
    4. Inverse Limits
    5. Tensor Products
    6. Properties of Tensor Products
    7. Dual Modules
    8. Flat Modules
    9. Completions

  • XII. Ext and Tor
    1. Complexes
    2. Resolutions
    3. Derived Functors
    4. Ext
    5. Tor
    6. Universal Coefficient Theorems
    7. Cohomology of Groups
    8. Projective Dimension
    9. Global Dimension

  • XIII. Algebras
    1. Algebras over a Ring
    2. The Tensor Algebra
    3. The Symmetric Algebra
    4. The Exterior Algebra
    5. Tensor Products of Algebras
    6. Tensor Products of Fields
    7. Simple Algebras over a Field

  • XIV.Lattices
    1. Definitions
    2. Complete Lattices
    3. Modular Lattices
    4. Distributive Lattices
    5. Boolean Lattices

  • XV.Universal Algebra
    1. Universal Algebras
    2. Word Algebras
    3. Varieties
    4. Subdirect Products

  • XVI.Categories
    1. Definition and Examples
    2. Functors
    3. Limits and Colimits
    4. Completeness
    *5. Additive Categories
    6. Adjoint Functors
    7. The Adjoint Functor Theorem
    8. Triples
    9. Tripleability
    10. Varieties

  • Appendix
    1. Chain Conditions
    2. The Axiom of Choice
    3. Ordinal Numbers
    4. Ordinal Induction
    5. Cardinal Numbers
    References
    Further Readings
    Index

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