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

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

Published $2003$, 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.$\quad$Qudratic 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