Book:P.M. Cohn/Basic Algebra: Groups, Rings and Fields

Subject Matter

 * Abstract Algebra

Contents

 * Preface


 * Conventions on Terminology


 * $1. \quad$ Sets
 * $1.1 \quad$ Finite, Countable and Uncountable Sets
 * $1.2 \quad$ Zorn's Lemma and Well-ordered Sets
 * $1.3 \quad$ Graphs


 * $2. \quad$ Groups
 * $2.1 \quad$ Definition and Basic Properties
 * $2.2 \quad$ Permutation Groups
 * $2.3 \quad$ The Isomorphism Theorems
 * $2.4 \quad$ Soluble and Nilpotent Groups
 * $2.5 \quad$ Commutators
 * $2.6 \quad$ The Frattini Subgroup and the Fitting Subgroup


 * $3. \quad$ Lattices and Categories
 * $3.1 \quad$ Definition; Modular and Distributive Lattices
 * $3.2 \quad$ Chain Conditions
 * $3.3 \quad$ Categories
 * $3.4 \quad$ Boolean Algebras


 * $4. \quad$ Rings and Modules
 * $4.1 \quad$ The Definitions Recalled
 * $4.2 \quad$ The Category of Modules over a Ring
 * $4.3 \quad$ Semisimple Modules
 * $4.4 \quad$ Matrix Rings
 * $4.5 \quad$ Direct Products of Rings
 * $4.6 \quad$ Free Modules
 * $4.7 \quad$ Projective and Injective Modules
 * $4.8 \quad$ The Tensor Product of Modules
 * $4.9 \quad$ Duality of Finite Abelian Groups


 * $5. \quad$ Algebras
 * $5.1 \quad$ Algebras; Definition and Examples
 * $5.2 \quad$ The Wedderburn Structure Theorems
 * $5.3 \quad$ The Radical
 * $5.4 \quad$ The Tensor Product of Algebras
 * $5.5 \quad$ The Regular Representation; Norm and Trace
 * $5.6 \quad$ Möbius Functions


 * $6. \quad$ Multilinear Algebra
 * $6.1 \quad$ Graded Algebras
 * $6.2 \quad$ Free Algebras and Tensor Algebras
 * $6.3 \quad$ The Hilbert Series of a Graded Ring or Module
 * $6.4 \quad$ The Exterior Algebra on a Module


 * $7. \quad$ Field Theory
 * $7.1 \quad$ Fields and their Extensions
 * $7.2 \quad$ Splitting Fields
 * $7.3 \quad$ The Algebraic Closure of a Field
 * $7.4 \quad$ Separability
 * $7.5 \quad$ Automorphisms of Field Extensions
 * $7.6 \quad$ The Fundamental Theorem of Galois Theory
 * $7.7 \quad$ Roots of Unity
 * $7.8 \quad$ Finite Fields
 * $7.9 \quad$ Primitive Elements; Norm and Trace
 * $7.10 \quad$ Galois Theory of Equations
 * $7.11 \quad$ The Solution of Equation by Radicals


 * $8. \quad$ Quadratic Forms and Ordered Fields
 * $8.1 \quad$ Inner Product Spaces
 * $8.2 \quad$ Orthogonal Sums and Diagonalization
 * $8.3 \quad$ The Orthogonal Group of a Space
 * $8.4 \quad$ The Clifford Algebra and the Spinor Norm
 * $8.5 \quad$ Witt's Cancellation Theorem and the Witt Group of a Field
 * $8.6 \quad$ Ordered Fields
 * $8.7 \quad$ The Field of Real Numbers
 * $8.8 \quad$ Formally Real Numbers
 * $8.9 \quad$ The Witt Ring of a Field
 * $8.10 \quad$ The Symplectic Group
 * $8.11 \quad$ Quadratic Forms in Characteristic Two


 * $9. \quad$ Valuation Theory
 * $9.1 \quad$ Divisibilty and Valuations
 * $9.2 \quad$ Absolute Values
 * $9.3 \quad$ The $p$-adic Numbers
 * $9.4 \quad$ Extensions of Valuations


 * $10. \quad$ Commutatve Rings
 * $10.1 \quad$ Operations on Ideals
 * $10.2 \quad$ Prime Ideals and Factorization
 * $10.3 \quad$ Localization
 * $10.4 \quad$ Noetherian Rings
 * $10.5 \quad$ Dedekind Domains
 * $10.6 \quad$ Modules over Dedekind Domains
 * $10.7 \quad$ Algebraic Equations
 * $10.8 \quad$ The Primary Decomposition
 * $10.9 \quad$ Dimension
 * $10.10 \quad$ The Hilbert Nullstellensatz


 * $11. \quad$ Infinite Field Extensions
 * $11.1 \quad$ Abstract Dependence Relations
 * $11.2 \quad$ Algebraic Dependence
 * $11.3 \quad$ Simple Trancendental Extensions
 * $11.4 \quad$ Separable and $p$-radical Extensions
 * $11.5 \quad$ Derivations
 * $11.6 \quad$ Linearly Disjoint Extensions
 * $11.7 \quad$ Composites of Fields
 * $11.8 \quad$ Infinite Algebraic Extensions
 * $11.9 \quad$ Galois Descent
 * $11.10 \quad$ Kummer Extensions


 * Bibliography


 * List of Notations


 * Author Index


 * Subject Index