# Book:P.M. Cohn/Algebra/Volume 2/Second Edition

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## P.M. Cohn:

## P.M. Cohn: *Algebra Volume 2, 2nd ed.*

Published $\text {1989}$, **Wiley**

- ISBN 0 471 92235 8.

### Subject Matter

### Contents

**Preface to the Second Edition**

**From the Preface to the First Edition**

**Conventions on terminology**

**Table of interdependence of chapters (Leitfaden)**

**1 Sets**- 1.1 Finite, countable and uncountable sets
- 1.2 Zorn's lemma and well-ordered sets
- 1.3 Categories
- 1.4 Graphs
- Further exercises

**2 Lattices**- 2.1 Definitlons, modular and distributive lattices
- 2.2 Chain conditions
- 2.3 Boolean algebras
- 2.4 Möbius functions
- Further exercises

**3 Field theory**- 3.1 Flelds and their extensions
- 3.2 Splitting fields
- 3.3 The algebraic closure of a field
- 3.4 Separability
- 3.5 Automorphisms of field extensions
- 3.6 The fundamental theorem of Galols theory
- 3.7 Roots of unity
- 3.8 Finite fields
- 3.9 Primitive elements; norm and trace
- 3.10 Galois theory of equations
- 3.11 The solution of equations by radicals
- Further exercises

**4 Modules**- 4.1 The category of modules over a ring
- 4.2 Semisimple modules
- 4.3 Matrix rings
- 4.4 Free modules
- 4.5 Projective and injective modules
- 4.6 Duality of finite abelian groups
- 4.7 The tensor product of modules
- Further exercises

**5 Rings and algebras**- 5.1 Algebras: definition and examples
- 5.2 Direct products of rings
- 5.3 The Wedderburn structure theorems
- 5.4 The radical
- 5.5 The tensor product of algebras
- 5.6 The regular representation; norm and trace
- 5.7 Composites of fields
- Further exercises

**6 Quadratic forms and ordered fields**- 6.1 Inner product spaces
- 6.2 Orthogonal sums and diagonalization
- 6.3 The orthogonal group of a space
- 6.4 Witt's cancellation theorem and the Witt group of a field
- 6.5 Ordered fields
- 6.6 The field of real numbers
- Further exercises

**7 Representation theory of finite groups**- 7.1 Basic definitions
- 7.2 The averaging lemma and Maschke's theorem
- 7.3 Orthogonality and completeness
- 7.4 Characters
- 7.5 Complex representations
- 7.6 Representations of the symmetric group
- 7.7 Induced representations
- 7.8 Applications: the theorems of Burnside and Frobenius
- Further exercises

**8 Valuation theory**- 8.1 Divisibility and valuations
- 8.2 Absolute values
- 8.3 The $p$-adic numbers
- 8.4 Integral elements
- 8.5 Extension of valuations
- Further exercises

**9 Commutative rings**- 9.1 Operations on ideals
- 9.2 Prime ideals and factorisation
- 9.3 Localisation
- 9.4 Noetherian rings
- 9.5 Dedekind domains
- 9.6 Modules over Dedekind domains
- 9.7 Algebraic equations
- 9.8 The primary decomposition
- 9.9 Dimension
- 9.10 The Hilbert Nullstellensatz
- Further exercises

**10 Coding theory**- 10.1 The transmission of information
- 10.2 Block codes
- 10.3 Linear codes
- 10.4 Cyclic codes
- 10.5 Other codes
- Further exercises

**11 Languages and automata**- 11.1 Monoids and monoid actions
- 11.2 Languages and grammars
- 11.3 Automata
- 11.4 Variable-length codes
- 11.5 Free algebras and formal power series rings
- Further exercises

- Bibliography

- List of notations

- Index