Book:R.B.J.T. Allenby/Rings, Fields and Groups: An Introduction to Abstract Algebra/Second Edition

Subject Matter

 * Abstract Algebra

Contents

 * Preface to the first edition
 * Preface to the second edition
 * How to read this book
 * Prologue


 * 0 Elementary set theory and methods of proof
 * 0.1 Introduction
 * 0.2 Sets
 * 0.3 New sets from old
 * 0.4 Some methods of proof


 * 1 Numbers and polynomials
 * 1.1 Introduction
 * 1.2 The basic axioms. Mathematical induction
 * 1.3 Divisibility, irreducibles and primes in $\Z$
 * Biography and portrait of Hilbert
 * 1.4 GCDs
 * 1.5 The unique factorisation theorem (two proofs)
 * 1.6 Polynomials&mdash;what are they?
 * 1.7 The basic axioms
 * 1.8 The 'new' notation
 * 1.9 Divisibility, irreducibles and primes in $\Q[x]$
 * 1.10 The division algorithm
 * 1.11 Roots and the remainder theorem


 * 2 Binary relations and binary operations
 * 2.1 Introduction
 * 2.2 Congruence mod $n$. Binary relations
 * 2.3 Equivalence relations and partitions
 * 2.4 $\Z_n$
 * Biography and portrait of Gauss
 * 2.5 Some deeper number-theoretic results concerning congruences
 * 2.6 Functions
 * 2.7 Binary operations


 * 3 Introduction to rings
 * 3.1 Introduction
 * 3.2 The abstract definition of a ring
 * Biography and portrait of Hamilton
 * 3.3 Ring properties deducible from the axioms
 * 3.4 Subrings, subfields and ideals
 * Biography and portrait of Noether
 * Biography and portrait of Fermat
 * 3.5 Fermat's conjecture (FC)
 * 3.6 Divisibility in rings
 * 3.7 Euclidean rings, unique factorisation domains and principal ideal domains
 * 3.8 Three number-theoretic applications
 * Biography and portrait of Dedekind
 * 3.9 Unique factorisation reestablished. Prime and maximal ideals
 * 3.10 Isomorphism. Fields of fractions. Prime subfields.
 * 3.11 $U[x]$ where $U$ is a UFD
 * 3.12 Ordered domains. The uniqueness of $\Z$


 * 4 Factor rings and fields
 * 4.1 Introduction
 * 4.2 Return to roots. Ring homomorphisms. Kronecker's theorem
 * 4.3 The isomorphism theorems
 * 4.4 Constructions of $\R$ from $\Q$ and of $\C$ from $\R$
 * Biography and portrait of Cauchy
 * 4.5 Finite fields
 * Biography and portrait of Moore
 * 4.6 Constructions with compass and straightedge
 * 4.7 Symmetric polynomials
 * 4.8 The fundamental theorem of algebra


 * 5 Basic group theory
 * 5.1 Introduction
 * 5.2 Beginnings
 * Biography and portrait of Lagrange
 * 5.3 Axioms and examples
 * 5.4 Deductions from the axioms
 * 5.5 The symmetric and the alternating groups
 * 5.6 Subgroups. The order of an element
 * 5.7 Cosets of subgroups. Lagrange's theorem
 * 5.8 Cyclic groups
 * 5.9 Isomorphism. Group tables
 * Biography and portrait of Cayley
 * 5.10 Homomorphisms. Normal subgroups
 * 5.11 Factor groups. The first isomorphism theorem
 * 5.12 Space groups and plane symmetry groups


 * 6 Structure theorems of group theory
 * 6.1 Introduction
 * 6.2 Normaliser. Centraliser. Sylow's theorems
 * 6.3 Direct products
 * 6.4 Finite abelian groups
 * 6.5 Soluble groups. Composition series
 * 6.6 Some simple groups


 * 7 A brief excursion into Galois theory
 * 7.1 Introduction
 * Biography and portrait of Galois
 * 7.2 Radical Towers and Splitting Fields
 * 7.3 Examples
 * 7.4 Some Galois groups: their orders and fixed fields
 * 7.5 Separability and Normability
 * 7.6 Subfields and subgroups
 * 7.7 The groups $\operatorname{Gal}({\rm R/F})$ and $\operatorname{Gal}({\rm S_f/F})$
 * 7.8 The groups $\operatorname{Gal}({\rm F_{i,j}/F_{i,j-1}})$
 * 7.9 A Necessary condition for the solubility of a polynomial equation by radicals
 * Biography and portrait of Abel
 * 7.10 A Sufficient condition for the solubility of a polynomial equation by radicals
 * 7.11 Non-soluble polynomials: grow your own!
 * 7.12 Galois Theory&mdash;old and new


 * Partial solutions to the exercises
 * Bibliography
 * Notation
 * Index