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

## R.B.J.T. Allenby: Rings, Fields and Groups: An Introduction to Abstract Algebra (2nd Edition)

Published $\text {1991}$, Edward Arnold

ISBN 0-7131-3476-3.

### Contents

Preface to the first edition
Preface to the second edition
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—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
4.3 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—old and new
Partial solutions to the exercises
Bibliography
Notation
Index