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.$\quad$Qudratic 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