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

Subject Matter

• Abstract Algebra

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