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

## P.M. Cohn: Algebra, Volume $\text { 1 }$ (2nd Edition)

Published $\text {1982}$, Wiley

ISBN 0 471 10169 9

### 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
8.1 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-HÃ¶lder 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