Rings, Fields and Groups. An Introduction to Abstract Algebra

Cover
Preface
Contents
How to read this book
Prologue
What is algebra?
Abstraction and axiomatisation
Historical development of algebra
Some uses of algebra
Problems
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
Exercises
1. Numbers and polynomials
1.1 Introduction
1.2 The basic axioms. Mathematical induction
Exercises
1.3 Divisibility, irreducibles and primes in Z
Exercises
1.4 GCDs
Exercises
1.5 The unique factorisation theorem (two proofs)
Exercises
1.6 Polynomials-what are they?
Exercises
1.7 The basic axioms
Exercises
1.8 The 'new' notation
Exercises
1.9 Divisibility, irreducibles and primes in O[x]
Exercises
1.10 The division algorithm
Exercises
1.11 Roots and the remainder theorem
Exercises
2. Binary relations and binary operations
2.1 Introduction
2.2 Congruence mod n. Binary relations
Exercises
2.3 Equivalence relations and partitions
Exercises
2.4 Z_n
Exercises
2.5 Some deeper number-theoretic results concerning congruences
Exercises
2.6 Functions
Exercises
2.7 Binary operations
Exercises
3. Introduction to rings
3.1 Introduction
3.2 The abstract definition of a ring
Exercises
3.3 Ring properties deducible from the axioms
Exercises
3.4 Subrings, subfields and ideals
Exercises
3.5 Fermat's conjecture (FC)
Exercises
3.6 Divisibility in rings
Exercises
3.7 Euclidean rings, unique factorisation domains and principal ideal domains
Exercises
3.8 Three number-theoretic applications
Exercises
3.9 Unique factorisation reestablished. Prime and maximal ideals
Exercises
3.10 Isomorphism. Fields of fractions. Prime subfields
Exercises
3.11 U[x] where U is a UFD
Exercises
3.12 Ordered domains. The uniqueness of Z
Exercises
4. Factor rings and fields
4.1 Introduction
4.2 Return to roots. Ring homomorphisms. Kronecker's theorem
Exercises
4.3 The isomorphism theorems
Exercises
4.4 Constructions of R from Q and of C from R
Exercises
4.5 Finite fields
Exercises
4.6 Constructions with compass and straightedge
Exercises
4.7 Symmetric polynomials
Exercises
4.8 The fundamental theorem of algebra
5. Basic group theory
5.1 Introduction
5.2 Beginnings
Exercises
5.3 Axioms and examples
Exercises
5.4 Deductions from the axioms
Exercises
5.5 The symmetric and the alternating groups
Exercises
5.6 Subgroups. The order of an element
Exercises
5.7 Cosets of subgroups. Lagrange's theorem
Exercises
5.8 Cyclic groups
Exercises
5.9 Isomorphism. Group tables
Exercises
5.10 Homomorphisms. Normal subgroups
Exercises
5.11 Factor groups. The first isomorphism theorem
Exercises
5.12 Space groups and plane symmetry groups
Exercise
6. Structure theorems of group theory
6.1 Introduction
6.2 Normaliser. Centraliser. Sylow's theorems
Exercises
6.3 Direct products
Exercises
6.4 Finite abelian groups
Exercises
6.5 Soluble groups. Composition series
Exercises
6.6 Some simple groups
Exercises
6.7 The theorem of Abel and Ruffini
Exercises
Epilogue
Bibliography
Algebra
Number theory
Logic, set theory
Applications
History
Papers
Other references
Notation
Index
Bavk Cover