# Book:P.M. Cohn/Basic Algebra: Groups, Rings and Fields

## P.M. Cohn: Basic Algebra: Groups, Rings and Fields

Published $\text {2003}$, Springer

ISBN 1-85233-587-4

### Contents

Preface
Conventions on Terminology
$1. \quad$ Sets
$1.1 \quad$ Finite, Countable and Uncountable Sets
$1.2 \quad$ Zorn's Lemma and Well-ordered Sets
$1.3 \quad$ Graphs
$2. \quad$ Groups
$2.1 \quad$ Definition and Basic Properties
$2.2 \quad$ Permutation Groups
$2.3 \quad$ The Isomorphism Theorems
$2.4 \quad$ Soluble and Nilpotent Groups
$2.5 \quad$ Commutators
$2.6 \quad$ The Frattini Subgroup and the Fitting Subgroup
$3. \quad$ Lattices and Categories
$3.1 \quad$ Definition; Modular and Distributive Lattices
$3.2 \quad$ Chain Conditions
$3.3 \quad$ Categories
$3.4 \quad$ Boolean Algebras
$4. \quad$ Rings and Modules
$4.1 \quad$ The Definitions Recalled
$4.2 \quad$ The Category of Modules over a Ring
$4.3 \quad$ Semisimple Modules
$4.4 \quad$ Matrix Rings
$4.5 \quad$ Direct Products of Rings
$4.6 \quad$ Free Modules
$4.7 \quad$ Projective and Injective Modules
$4.8 \quad$ The Tensor Product of Modules
$4.9 \quad$ Duality of Finite Abelian Groups
$5. \quad$ Algebras
$5.1 \quad$ Algebras; Definition and Examples
$5.2 \quad$ The Wedderburn Structure Theorems
$5.3 \quad$ The Radical
$5.4 \quad$ The Tensor Product of Algebras
$5.5 \quad$ The Regular Representation; Norm and Trace
$5.6 \quad$ Möbius Functions
$6. \quad$ Multilinear Algebra
$6.1 \quad$ Graded Algebras
$6.2 \quad$ Free Algebras and Tensor Algebras
$6.3 \quad$ The Hilbert Series of a Graded Ring or Module
$6.4 \quad$ The Exterior Algebra on a Module
$7. \quad$ Field Theory
$7.1 \quad$ Fields and their Extensions
$7.2 \quad$ Splitting Fields
$7.3 \quad$ The Algebraic Closure of a Field
$7.4 \quad$ Separability
$7.5 \quad$ Automorphisms of Field Extensions
$7.6 \quad$ The Fundamental Theorem of Galois Theory
$7.7 \quad$ Roots of Unity
$7.8 \quad$ Finite Fields
$7.9 \quad$ Primitive Elements; Norm and Trace
$7.10 \quad$ Galois Theory of Equations
$7.11 \quad$ The Solution of Equation by Radicals
$8. \quad$ Quadratic Forms and Ordered Fields
$8.1 \quad$ Inner Product Spaces
$8.2 \quad$ Orthogonal Sums and Diagonalization
$8.3 \quad$ The Orthogonal Group of a Space
$8.4 \quad$ The Clifford Algebra and the Spinor Norm
$8.5 \quad$ Witt's Cancellation Theorem and the Witt Group of a Field
$8.6 \quad$ Ordered Fields
$8.7 \quad$ The Field of Real Numbers
$8.8 \quad$ Formally Real Numbers
$8.9 \quad$ The Witt Ring of a Field
$8.10 \quad$ The Symplectic Group
$8.11 \quad$ Quadratic Forms in Characteristic Two
$9. \quad$ Valuation Theory
$9.1 \quad$ Divisibilty and Valuations
$9.2 \quad$ Absolute Values
$9.3 \quad$ The $p$-adic Numbers
$9.4 \quad$ Extensions of Valuations
$10. \quad$ Commutatve Rings
$10.1 \quad$ Operations on Ideals
$10.2 \quad$ Prime Ideals and Factorization
$10.3 \quad$ Localization
$10.4 \quad$ Noetherian Rings
$10.5 \quad$ Dedekind Domains
$10.6 \quad$ Modules over Dedekind Domains
$10.7 \quad$ Algebraic Equations
$10.8 \quad$ The Primary Decomposition
$10.9 \quad$ Dimension
$10.10 \quad$ The Hilbert Nullstellensatz
$11. \quad$ Infinite Field Extensions
$11.1 \quad$ Abstract Dependence Relations
$11.2 \quad$ Algebraic Dependence
$11.3 \quad$ Simple Trancendental Extensions
$11.4 \quad$ Separable and $p$-radical Extensions
$11.5 \quad$ Derivations
$11.6 \quad$ Linearly Disjoint Extensions
$11.7 \quad$ Composites of Fields
$11.8 \quad$ Infinite Algebraic Extensions
$11.9 \quad$ Galois Descent
$11.10 \quad$ Kummer Extensions
Bibliography
List of Notations
Author Index
Subject Index