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

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## R.B.J.T. Allenby:

## 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.

### Subject Matter

### 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—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—old and new

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