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

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

Next

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.