Book:P.M. Cohn/Algebra/Volume 1/Second Edition

Subject Matter

 * Abstract Algebra

Contents

 * Preface to Second Edition


 * From the Preface to First Edition


 * Table of interdependence of chapters


 * 1 Sets and mappings
 * 1.1 The need for logic
 * 1.2 Sets
 * 1.3 Mappings
 * 1.4 Equivalence relations
 * 1.5 Ordered sets
 * Further exercises


 * 2 Integers and rational numbers
 * 2.1 The integers
 * 2.2 Divisibility and factorization in $\mathbf Z$
 * 2.3 Congruences
 * 2.4 The rational numbers and some finite fields
 * Further exercises


 * 3 Groups
 * 3.1 Monoids
 * 3.2 Groups; the axioms
 * 3.3 Group actions and coset decompositions
 * 3.4 Cyclic groups
 * 3.5 Permutation groups
 * 3.6 Symmetry
 * Further exercises


 * 4 Vector spaces and linear mappings
 * 4.1 Vectors and linear dependence
 * 4.2 Linear mappings
 * 4.3 Bases and dimension
 * 4.4 Direct sums and quotient spaces
 * 4.5 The space of linear mappings
 * 4.6 Change of basis
 * 4.7 The rank
 * 4.8 Affine spaces
 * 4.9 Category and functor
 * Further exercises


 * 5 Linear equations
 * 5.1 Systems of linear equations
 * 5.2 Elementary operations
 * 5.3 Linear programming
 * 5.4 $PAQ$-reduction and the inversion of matrices
 * 5.5 Block multiplication
 * Further exercises


 * 6 Rings and fields
 * 6.1 Definitions and examples
 * 6.2 The field of fractions of an integral domain
 * 6.3 The characteristic
 * 6.4 Polynomials
 * 6.5 Factorization
 * 6.6 The zeros of polynomials
 * 6.7 The factorization of polynomials
 * 6.8 Derivatives
 * 6.9 Symmetric and alternating functions
 * Further exercises


 * 7 Determinants
 * 7.1 Definition and basic properties
 * 7.2 Expansion of a determinant
 * 7.3 The determinantal rank
 * 7.4 The resultant
 * Further exercises


 * 8 Quadratic forms
 * 8.1 Bilinear forms and pairings
 * 8.2 Dual spaces
 * 8.3 Inner products; quadratic and hermitian forms
 * 8.4 Euclidean and unitary spaces
 * 8.5 Orthogonal and unitary matrices
 * 8.6 Alternating forms
 * Further exercises


 * 9 Further group theory
 * 9.1 The isomorphism theorems
 * 9.2 The Jordan-Hölder theorem
 * 9.3 Groups with operators
 * 9.4 Automorphisms
 * 9.5 The derived group; soluble groups and simple groups
 * 9.6 Direct products
 * 9.7 Abelian groups
 * 9.8 The Sylow theorems
 * 9.9 Generators and defining relations; free groups
 * Further exercises


 * 10 Rings and modules
 * 10.1 Ideals and quotient rings
 * 10.2 Modules over a ring
 * 10.3 Direct products and direct sums
 * 10.4 Free modules
 * 10.5 Principal Ideal domains
 * 10.6 Modules over a principal ideal domain
 * Further exercises


 * 11 Normal forms for matrices
 * 11.1 Eigenvalues and eigenvectors
 * 11.2 The $k \left[{x}\right]$-module defined by an endomorphism
 * 11.3 Cyclic endomorphisms
 * 11.4 The Jordan normal form
 * 11.5 The Jordan normal form: another method
 * 11.6 Normal matrices
 * 11.7 Linear algebras
 * Further exercises


 * Solutions to the exercises


 * Appendices
 * 1 Further reading
 * 2 Some frequently used notations


 * Index



Source work progress
* : $\S 3.4$: Cyclic groups: Exercise $9$


 * Second pass through:


 * : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Exercise $2$


 * Needs another pass through, full rigour not yet achieved.