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

Subject Matter

 * Abstract Algebra

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