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

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## P.M. Cohn:

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

Published $\text {1982}$, **Wiley**

- ISBN 0 471 10169 9.

### Subject Matter

### 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-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**- 1 Further reading
- 2 Some frequently used notations

**Index**

## Source work progress

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups: Exercise $9$

- Second pass through:

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): 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.