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

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

Published $\text {1989}$, Wiley

ISBN 0 471 92235 8

### 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 MÃ¶bius 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
5.5 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