Rings, Fields and Groups' gives a stimulating and unusual introduction to the results, methods and ideas now commonly studied on abstract algebra courses at undergraduate level. The author provides a mixture of informal and formal material which help to stimulate the enthusiasm of the student, whilst still providing the essential theoretical concepts necessary for serious study.
Retaining the highly readable style of its predecessor, this second edition has also been thoroughly revised to include a new chapter on Galois theory plus hints and solutions to many of the 800 exercises featured.
Readership: Undergraduate students of pure mathematics.
Author(s): Reg Allenby
Edition: 2nd
Publisher: Edward Arnold / Butterworth-Heinemann
Year: 1991
Language: English
Pages: xxvi+383
Cover
Rings,Fields and Groups: An Introduction to Abstract Algebra
Copyright
© 1991 R. B. J. T. Allenby
ISBN 0-7131-3476-3
Preface to the first edition
Contents
Preface to the second edition
How to read this book
Prologue
What is algebra?
Abstraction and axiomatisation
Historical development of algebra
Some uses of algebra
0 Elementary set theory and methods of proof
01 Introduction
0.2 Sets
03 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
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 Zn
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
3.3 Ring properties deducible from the axioms
3.4 Subrings, subfields and ideals
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
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
4.5 Finite fields
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
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
5.10 Homomorphisms. Normal subgroups
5.11 Factor groups. The first isomorphism theorem
512 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 GaloisTheory
7.1 Introduction
7.2 Radical towers and splitting fields
7.3 Examples
7.4 Some Galois groups: their orders and fixed fields
7.5 Separability and normality
7.6 Subfields and subgroups
7.7 The groups GaI(R/F) and GaI(Sf/F)
7.8 The groups GaI(F_i,j/F_i,j-1)
7.9 A necessary condition for the solubility of a polynomial equation by radicals
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
Exercises in Chapter 0
Exercises 1.2
Exercises 1.3, 1.4
Exercises 1.5
Exercises 1.6, 1.7, 1.8, 1.9
Exercises 1.10
Exercises 1.11, Exercises 2.2
Exercises 2.3, 2.4, 2.5
Exercises 2.6
Exercises 2.7, 3.2
Exercises 3.3
Exercises 3.4
Exercises 3.5, 3.6
Exercises 3.7
Exercises 3.8, 3.9
Exercises 3.10
Exercises 3.11
Exercises 3.12, 4.2
Exercises 4.3
Exercises 4.4
Exercises 4.5
Exercises 4.6
Exercises 4.7
Exercises 5.2, 5.3
Exercises 5.4, 5.5
Exercises 5.6
Exercises 5.7
Exercises 5.8
Exercises 5.9
Exercises 5.10
Exercises 5.11
Exercises 6.2
Exercises 6.3
Exercises 6.4
Exercises 6.5
Exercises 6.6
Exercises 72
Exercises 7.3
Exercises 7.4
Exercises 7.5
Exercises 7.6
Exercises 7.7
Exercises 7.8
Exercises 7.9
Exercises 7.10
Exercises 7.11
Exercises 7.12
Bibliography
Algebra
Number theory (including Number Systems)
Logic, set theory
Applications (in a broad sense)
History
Papers
Other references
Notation
Index
