1.1. From the Preface.

Amid all the current interest in modern algebra, field theory has been rather neglected - most of the recent textbooks in algebra have been concerned with groups or vector spaces. But field theory is a very attractive branch of algebra, with many fascinating applications; and its central result, the Fundamental Theorem of Galois Theory, is by any standards one of the really "big" theorems of mathematics. This book aims to bring the reader from the basic definitions to important results and to introduce him to the spirit and some of the techniques of abstract algebra. It presupposes only a little knowledge of elementary group theory and a willingness on the reader's part to remember definitions precisely and to engage in close argument.

Chapter I develops ab initio the elementary properties of rings, fields and vector spaces. Chapter 2 describes extensions of fields and various ways of classifying them. In Chapter 3 we give an exposition of the Galois theory of normal separable extensions of finite degree, closely following Artin's approach. Chapter 4 provides a wide variety of applications of the preceding theory, including the classification of all fields with a finite number of elements, ruler-and-compasses constructions and the impossibility of solving by radicals the generic polynomial of degree greater than 4.

I had the great good fortune to persuade my colleague Dr Hamish Anderson to read the first draft of this book, and as a result of his careful scrutiny and penetrating comments many blemishes were removed; I am deeply grateful to him for his invaluable help. ... It is a pleasure also to record my great gratitude to Professor D E Rutherford, whose lectures on groups first aroused my interest in algebra, for his constant encouragement and help at all stages in the preparation of the book. Finally I must not forget to thank several generations of honours students in The Queen's University of Belfast and Queen's College, Dundee who patiently listened to and commented on the successive lecture courses which eventually turned into this book; one of them in particular, in a spontaneous exclamation, provided me with an appropriate conclusion to Chapter 4.

1.2. Contents.

Preface.

Part I: Elementary Definitions.

1. Rings and fields;
2. Elementary properties;
3. Homomorphisms;
4. Vector spaces;
5. Polynomials;
6. Higher polynomial rings; rational functions.

Part II: Extensions of fields.

7. Elementary properties;
8. Simple extensions;
9. Algebraic extensions;
10. Factorisation of polynomials;
11. Splitting fields;
12. Algebraically closed fields;
13. Separable extensions.

Part III: Galois theory.

14. Automorphisms of fields;
15. Normal extensions;
16. The fundamental theorem of Galois Theory;
17. Norms and traces;
18. The primitive element theorem; Lagrange's theorem;
19. Normal bases.

Part IV: Applications.

20. Finite fields;
21. Cyclotomic extensions;
22. Cyclotomic extensions of the rational number field;
23. Cyclic extensions;
24. Wedderburn's theorem;
25. Ruler-an-compasses constructions;
26. Solution by radicals;
27. Generic polynomials.

1.3. Review by: S Crampe-Priess.
Mathematical Reviews MR0199176 (33 #7325).

The book is a very clearly written introduction to the algebraic theory of (commutative) fields. Only the elementary theory of finite groups is required as knowledge. The Galois theory of finite separable normal extensions is presented in detail (according to Artin). There are numerous illustrative examples and a number of exercises at the end of each chapter.

The book consists of four chapters: In Chapter I the elementary properties of rings, fields and vector spaces are developed. Chapter II deals with extensions of fields. Chapter III presents the Galois theory of finite separable normal extensions, and Chapter IV brings numerous applications of the previous theory, such as the classification of finite fields, the problems of "constructions with compass and straight-edge" (squaring the circle, doubling the cube, trisection of the angle) and the insolubility by radicals of the general $n$-th degree equation over a field of characteristic 0 for $n ≥ 5$.

1.4. Review by: Neil Grabois.
The American Mathematical Monthly 73 (9) (1966), 1033.

This is an attractive book on field theory and Galois theory at a level somewhat beyond that of Artin's Galois Theory. Some knowledge of group theory is assumed and the book is addressed to honours and graduate students. The fundamental theorem of finitely generated abelian groups and some elementary facts about prime power order groups are freely used.

There are four chapters: Chapter 1 introduces rings, fields and vector spaces as well as developing the basic facts about them; Chapter 2 is devoted to extensions of fields, with particular care given to field identifications; the basic theorems concerning the division algorithm and the existence of algebraic closures are stated without proofs; Chapter 3 is devoted to the Galois theory of finite dimensional Galois extensions and mirrors Artin's approach; Chapter 4 offers applications of the previous material and includes material on cyclotomic and cyclic extensions, Wedderburn's theorem on finite division rings, an especially complete discussion of constructability by compass and straight-edge, and the impossibility of solving by radicals the general equation of degree > 4.

The text is very clearly written, with many examples, and the exercises are good though there are too few for my taste. All told this is an excellent introduction to field theory.

1.5. Review by: D B Scott.
The Mathematical Gazette 49 (370) (1965), 466.

The first paragraph of the author's preface provides a very fair and realistic appraisal of the work. The presentation is clear and convincing. It is perhaps a hard-headed rather than an exciting presentation, but the author, an ardent Scotsman writing in the Edinburgh series, would presumably wish to have it that way. Just as the first paragraph of the Preface provides the most effective review so the next paragraph gives the best description of the contents ...

2.1. From the Preface.

This book is intended to provide an introduction to the basic facts about modules, abelian categories and homological algebra and to apply these in deriving the classical results on Artinian rings and simple and separable algebras. I hope it may be found to give a clear, connected and not over-condensed account of some more or less well-established topics in ring theory from which the reader may proceed without too much difficulty to study the current research work into the structure of rings, category theory and homological algebra. Apart from a few results on fields, for which the reader may refer to my Introduction to Field Theory, the exposition is entirely self-contained. In Chapter 1 we discuss the elementary ideas of rings, ideals, homomorphisms and extensions of rings.

Chapter 2 begins with two sections on modules and their homomorphisms. These are followed by a study of certain important constructions - groups of homomorphisms, direct sums and products, direct and inverse limits and tensor products - and of free, projective and injective modules. The treatment of these topics, with its emphasis on universal and co-universal properties, is intended to provide a foundation for category theory. The chapter concludes with sections on composition series and Artinian and Noetherian modules and rings. Chapters 1 and 2 are expanded versions of Chapters 1 and 2 of my Elementary rings and modules.

Drawing on Chapter 2 for illustrative examples, Chapter 3 introduces the notions of category and functor, with special emphasis on abelian categories. It includes an exposition of Puppe's theory of relations in an abelian category, as first presented in his paper 'Korrespondenzen in abelsche Kategorien', Math. Ann. 148 (1962).

Chapter 4 presents an account of semisimple and simple modules and the Artin-Wedderburn structure theory for semisimple and simple rings. The last section of this chapter discusses the radical and includes a proof of Hopkins's theorem. Chapter 5 is concerned mainly with the theory of finite-dimensional simple algebras, including the Brauer group, splitting fields and the Skolem-Noether theorem. The chapter also contains a discussion of separable algebras.

The final chapter develops homological algebra in locally small abelian categories, proceeding as far as the construction of derived functors. It then specialises to the cohomology theory of algebras and the characterisation of algebras of dimension zero, concluding with a section on extensions of algebras and the Wedderburn Principal Theorem.

Most of the writing of this book was done while I was on leave from Dundee and I am happy to express my thanks to the University of Western Australia for affording me the hospitality of its beautiful campus and the stimulation of working in its friendly and lively department of Mathematics. Since returning to Dundee I have had the valuable assistance of Dr Arthur Sands, who read the whole manuscript, and of Dr Hamish Anderson, who read the proofs with great care; I am deeply indebted to them both. Finally I should like to express my gratitude to my wife who has borne with great patience the long period of gestation of this book.

2.2. Review by: H H Storrer.
Mathematical Reviews MR0332850 (48 #11175).

As stated in the author's preface, this book is intended to provide an introduction to the basic facts about modules, abelian categories and homological algebra and to apply these in deriving the classical results on Artinian rings and simple and separable algebras. The book is divided into six chapters. Chapter 1 begins with some general remarks on laws of composition and continues with the study of the elementary properties of rings. Modules are introduced in Chapter 2, together with a number of important concepts such as groups of homomorphisms, direct sums and products, tensor products, free, projective, injective and flat modules, the Jordan-Hölder theorem, Artinian and Noetherian modules and rings and the Hilbert basis theorem. A notable feature is the treatment of direct and inverse limits, a topic not usually found in an introductory text. ...
...
The book is virtually self-contained, except for a few facts from field theory, needed in Chapter 5. The presentation is very clear, detailed and complete, only rarely a proof or even a small argument is left to the reader. This should be particularly useful for students reading the book on their own. Perhaps they will miss exercises; the book contains none. A list comprising 17 books and one research paper is intended as a guide to further reading; besides this, there is no bibliography.

3.1. Review by: Editors.
Mathematical Review MR0345993 (49 #10719).

The first two chapters of this introductory text (Rings and ideals, pp. 1-31; Modules, pp. 32-93) are essentially the more elementary sections of the author's Rings, modules and algebras [Oliver and Boyd, Edinburgh, 1971]. The third chapter (commutative rings, pp. 94-133) is devoted to topics that are fundamental to algebraic number theory and algebraic geometry.

3.2. Review by: Colin R Fletcher.
The Mathematical Gazette 57 (400) (1973), 145.

To start right at the beginning, and one cannot get further back than the title, it should perhaps be made clear for the wary that the word "elementary" is non-mathematical. The book consists of three chapters giving the elementary properties of rings (in general), of modules (this chapter taking up practically half the book), and commutative rings (unique factorisation and Dedekind domains, integral dependence and rings of fractions). All fundamental stuff, but is another book on abstract algebra really necessary? The onus must be on the author to show that it is. Van der Waerden, in the preface to Science Awakening, has a section entitled "What is new in this book". It would be asking too much to insist that all books should contain material which has not previously been published, or which was only available in mathematical journals. On the other hand a section entitled "Why I wrote this book" or better "Why anyone should bother to buy this book," perhaps should be obligatory and certainly would be illuminating.

I must own to the fact that I enjoyed reading the work. The author's style makes the pages and the time fly. Any lecturer needing a textbook for Hom, $\otimes$, projective and injective modules, direct sums and products need look no further. The only snag is this word "elementary". All sequences are exact and so no homology groups appear. Most exact sequences are short and there is no hint of homological dimension. The Schanuel lemma is not introduced, even as an exercise. For all these (and more) a new text will be needed.

Misprints are rare, mathematical errors are rarer (according to the definition 0 is irreducible in a field but not in a non-field). I only found one exercise which was not do-able (but I did not attempt them all). A handful of objects were not defined, ranging from non-commutative (not commutative or not necessarily commutative?) through commutative diagram to $R[x]$. Some results are quoted in passing (e.g. the division and Euclidean algorithms) which is fair enough, but to act similarly with Zorn's lemma is not. The statement of this axiom should be written out formally.

Finally, two personal bees in the personal bonnet. After Motzkin's work there is no justification for including the condition $\nu (ab) ≥ \nu (a)$ in the definition of a Euclidean domain. And if the proofs of the connections between Euclidean domain, principal ideal domain, etc., etc. are scattered through the text then the collocation of these results, possibly in the form of a diagram, should appear somewhere, even if only in the exercises, to wrap up the complete bundle. And to tie everything together the inverse implications should be disproved or at least stated not to hold.

However, these are minor criticisms. This is another useful addition to the Oliver and Boyd series and as an elementary introduction to the topics of rings and modules it is to be recommended.

4.1. From the Publisher.

A fascinating branch of algebra with numerous applications, field theory leads the way to one of the most important theorems of mathematics, the fundamental theorem of Galois theory. This text ranges from field theory's basic definitions to its most significant results and applications, introducing both the spirit and techniques of abstract algebra. Appropriate for undergraduate students in pure mathematics, it presupposes minimal knowledge of elementary group theory.

Acclaimed by American Mathematical Monthly as "an excellent introduction," this treatment begins by developing the elementary properties of rings and fields and examining a variety of homomorphisms, vector spaces, and polynomials. Subsequent chapters explore extension fields and their classifications as well as Artin's approach to Galois theory. The concluding section focuses on applications, with considerations of finite fields, cyclotomic extensions, solution by radicals, generic polynomials, and much more.

4.2. Review by: Editors.
Mathematical Reviews MR0345993 (49 #10719).

This appears to be a photographic reproduction of the first edition [Oliver & Boyd, Edinburgh, 1964], except for the change in publisher.

5.1. Review by: Victor Bryant.
The Mathematical Gazette 61 (416) (1977), 150.

The author's aim in this book is to present a course in elementary analysis using "set-theoretic concepts". By that he means not only clear definitions of functions and their domains, but also definitions of continuity etc. in terms of neighbourhoods. There is nothing wrong with this approach, and even at an elementary level I have seen some very successful texts along these lines. But I feel the current work fails on two important points. First, the author uses notation like $f:x \mapsto x^{2} (x \in R)$ for a function. This is not uncommon, and it does lead to a good understanding of functions and the overall ideas of differentiation and integration. However, the notation becomes extremely cumbersome to maintain throughout a study of the techniques of calculus, and I think that most mathematicians agree that the student of this notation must be weaned onto the traditional Leibniz notation in time for, say, integration by parts. This current work will do nothing to change their opinion. Dr Adamson's book is like one written in the i.t.a.: it is unpredictable how much a reader of it will understand of other books.

The Open University has used the 'arrow' notation for some years, and owing to the difficulties mentioned above it is to some extent phasing out the notation when it revises the foundation course in 1978. Dr Adamson avoids some of the difficulties by introducing special symbols for the functions $x \mapsto 0, x \mapsto 1$ and $x \mapsto x$; but this leads me to my second criticism. I do not feel that a long list of symbols unique to one particular text is very worthwhile. Even if the reader remembers them all, they are no use to him afterwards. ... Another disadvantage of a book with its own special notation is that it cannot usually be used as a reference book by students using another main text.

This work seems to have been produced by a photographic process, as specks of dust have been reproduced as clearly as the text. This is disconcerting in three ways: first I tried to brush away non-existent specks, secondly on one occasion I read an $h$ as a $k$, and thirdly when I had eventually conditioned myself to ignore the extra marks I found myself reading $h'$ as $h$.

The book covers all the material of a standard elementary rigorous analysis course from sequences and series to integration, and it contains plenty of exercises and examples. If I have been harsh, my criticism is of the approach and not of the way in which Dr Adamson has presented it. Someone who wished to prepare an analysis course in the style I have outlined would find this work of benefit.

6.1. From the Publisher.

All young computer scientists who aspire to write programs must learn something about algorithms and data structures. This book does exactly that. Based on lecture courses developed by the author over a number of years the book is written in an informal and friendly way specifically to appeal to students.

The book is divided into four parts: the first on Data Structures introduces a variety of structures and the fundamental operations associated with them, together with descriptions of how they are implemented in Pascal; the second discusses algorithms and the notion of complexity; Part III is concerned with the description of successively more elaborate structures for the storage of records and algorithms for retrieving a record from such a structure by means of its key; and finally, Part IV consists of very full solutions to nearly all the exercises in the book.

6.2. From the Preface.

In 1976 Niklaus Wirth, the inventor of Pascal, published a book entitled Algorithms + Data Structures = Programs. If the assertion of Wirth's title is correct - and it would be hard to dispute it - all young computer scientists who aspire to write programs must learn something about algorithms and data structures. This book is intended to help them do that. It is based on lecture courses developed over the past few years and I hope that at least some of the informality of the classroom and the spoken word has been transferred to the printed page. The lectures were given to first and second year students in The University of Dundee who had been well-grounded in Pascal and who had therefore already met some elementary data structures and sorting and searching algorithms; but only the syntax of Pascal was taken for granted, as it is in the book. My students had rather varied mathematical backgrounds and some were not very well-disposed to old-fashioned algebraic manipulation. A little of this (and a brief mention of limits) does appear in the book; but readers of a book are more fortunate than students in a classroom - they can skip all the details and concentrate on the final results.

6.3. Contents.

I Data structures

1 Arrays records and linked lists
1.1. Arrays
1.2. Storage of arrays
1.3. Records
1.4. Linked lists
1.5. Exercises 1

2. Stacks and queues

2.1. Stacks
2.2. Applications of stacks
2.3. Queues
2.4. Exercises 2

3. Binary trees

3.1. Binary trees
3.2. Binary search trees
3.3. Exercises 3

4. Heaps

4.1. Priority queues and heaps
4.2. Exercises 4

5. Graphs

5.1. Graphs and their implementation
5.2. Graphs and transversals
5.3. Exercises 5

II Algorithms

6. Algorithms and complexity

6.1. Algorithms
6.2. Complexity of algorithms
6.3. Exercises 6

7. Sorting algorithms

7.1. Internal sorting by comparisons
7.2. Other internal sorting algorithms
7.3. External sorting algorithms
7.4. Exercises 7

8. Graph algorithms

8.1. Shortest path algorithms
8.2. Spanning tree algorithms
8.3. Exercises 8

9. Some miscellaneous algorithms

9.1. Numerical multiplication algorithms
9.2. Matrix multiplication algorithms
9.3. A stable marriage algorithm
9.4. Exercises 9

III Storing and searching

10. Storing in arrays and lists

10.1. Sequential and binary searching
10.2. Hashing
10.3. Exercises 10

11. Storing in binary trees

11.1. Storing in binary search trees
11.2. Storing in AVL-trees
11.3. Exercises 11

12. Storing in multiway trees

12.1. Multiway search trees
12.2. B-trees
12.3. Tries
12.4. Exercises 12

IV Solutions

13. Solutions to Exercises 1

14. Solutions to Exercises 2

15. Solutions to Exercises 3

16. Solutions to Exercises 4

17. Solutions to Exercises 5

18. Solutions to Exercises 6

19. Solutions to Exercises 7

20. Solutions to Exercises 8

21. Solutions to Exercises 9

22. Solutions to Exercises 10

33. Solutions to Exercises 11

24. Solutions to Exercises 12

Index

7.1. From the Introduction.

This book has been called a Workbook to make it clear from the start that it is not a conventional textbook. Conventional textbooks proceed by giving in each section or chapter first the definitions of the terms to be used, the concepts they are to work with, then some theorems involving these terms (complete with proofs) and finally some examples and exercises to test the readers' understanding of the definitions and the theorems. Readers of this book will indeed find all the conventional constituents - definitions, theorems, proofs. examples and exercises but not in the conventional arrangement.

In the first part of the book will be found a quick review of the basic definitions of general topology interspersed with a large number of exercises, some of which are also described as theorems. (The use of the word Theorem is not intended as an indication of difficulty but of importance and usefulness.) The exercises are deliberately not "graded" - after all the problems we meet in mathematical "real life" do not come in order of difficulty; some of them are very simple illustrative examples; others are in the nature of tutorial problems for a conventional course, while others are quite difficult results. No solutions of the exercises, no proofs of the theorems are included in the first part of the book - this is a Workbook and readers are invited to try their hand at solving the problems and proving the theorems for themselves. I have been persuaded, with some reluctance, to offer suggestions about how to tackle the exercises which are not entirely straightforward; really dedicated Workbook-ers should ignore these! The second part of the book contains complete solutions to all but the most utterly trivial exercises and complete proofs of the theorems.

It has been widely recognised that general topology is a branch of mathematics particularly well adapted to independent study based on material carefully prepared by a teacher who otherwise gives minimal assistance. The most celebrated practitioner of this method was R L Moore (1882-1974) whose success with it is legendary. One factor in Moore's success must have been his insistence on hand-picking his students, preferring those who came to him tabula rasa. Few of his imitators and followers have enjoyed the luxury of being able to do this and have had to adapt his method to their more mundane circumstances. This book has grown from my attempts to provide a self-learning introduction to general topology for several generations of students in The University of Dundee, not all of whom would have been selected by Moore - though all of them responded enthusiastically to the method.

My Dundee students were presented, by instalments, with the introductory material and the exercises (though with much less generous hints than appear in the book) and were given the solutions only after the class had met to discuss the problems. Not all the students solved every problem for themselves, but there was a much higher participation rate than in conventional lecture courses. Two students who tackled the course by virtually unsupervised reading had particularly marked success.

In a class situation it is easy to ensure that students do not see the solutions to the exercises before they have tried to solve them; it is not so easy when exercises and solutions appear between the same two covers. But my readers are reminded that this is a Workbook and they are warmly invited to work at the exercises before turning to the solutions.

I must put on record my thanks to the Dundee students who have worked so cheerfully with the contents of this book, especially Malcolm Dobson and Ross Anderson. I am also deeply indebted to my friend Keith Edwards for his meticulous assistance, both typographical and mathematical. My gratitude to my wife for her encouragement during the writing and typesetting of this book is beyond measure.

7.2. Contents.

I Exercises.

- 1 Topological Spaces.
- 2 Mappings of Topological Spaces.
- 3 Induced and Co-induced Topologies.
- 4 Convergence.
- 5 Separation Axioms.
- 6 Compactness.
- 7 Connectedness.

II Answers.

- 8 Answers for Chapter 1.
- 9 Answers for Chapter 2.
- 10 Answers for Chapter 3.
- 11 Answers for Chapter 4.
- 12 Answers for Chapter 5.
- 13 Answers for Chapter 6.
- 14 Answers for Chapter 7.

- Further Reading.

8.1. From the Preface.

This book is a companion to A general topology workbook published by Birkhäuser last year. In an ideal world the order of publication would have been reversed, for the notation and some of the results of the present book are used in the topology book and on the other hand (the reader may be assured) no topology is used here.

Both books share the word Workbook in their titles. They are based on the principle that for at least some branches of mathematics a good way for a student to learn is to be presented with a clear statement of the definitions of the terms with which the subject is concerned and then to be faced with a collection of problems involving the terms just defined. In adopting this approach with my Dundee students of set theory and general topology I found it best not to differentiate too precisely between simple illustrative examples, easy exercises and results which in conventional textbooks would be labelled as Theorems. ... And, although readers may not find the hints I have provided in this book very generous, the students in my classes got none at all. Not only that, they were not shown the solutions to the exercises until they had made serious attempts to solve them for themselves. That was easy to arrange in the classroom situation but readers of the book can cheat by looking at the full answers provided in Part II. I should like to encourage them not to do that - and to remind them that this is a Workbook and that they will derive more benefit from it if they work seriously at the exercises before turning to the solutions.

The titles of Chapters 2 to 13 show that the contents of the book are entirely conventional for an advanced undergraduate or beginning graduate course in set theory. The treatment is intended to be relatively informal, in the sense that definitions and proofs are given for the most part in (hopefully clear) standard "mathematical English" not in logical symbolism. I hope that the reference in the Introduction to "the first order predicate calculus with equality" will not be found off-putting - it is, after all, only a shorthand way of describing the logical framework that we use as soon as we start to do mathematics seriously.

This is the fourth book in which I have had the help and advice of Nick Dawes with my Latex typesetting; as he breathes a sigh of relief that this is (I think) the last, I want to put on record my gratitude to him for his calm and calming expert assistance. I must again thank the several generations of students in The University of Dundee who wrestled successfully with the material presented here and whose success with it suggested that I turn it into this book. I am very grateful also to my friends - and particularly my wife and daughter - who encouraged me to complete it.

8.2. Contents.

I Exercises

1 First Axioms of the Theory NBG
2 Relations
3 Functional Relations and Mappings
4 Families of Sets
5 Equivalence Relations
6 Order Relations
7 Well-Ordering
8 Ordinals
9 Natural Numbers
10 Equivalents of the Axiom of Choice
11 Infinite Sets
12 Cardinals
13 Cardinal and Ordinal Arithmetic

II Answers

14 Answers to Chapter 1
15 Answers to Chapter 2
16 Answers to Chapter 3
17 Answers to Chapter 4
18 Answers to Chapter 5
19 Answers to Chapter 6
20 Answers to Chapter 7
21 Answers to Chapter 8
22 Answers to Chapter 9
23 Answers to Chapter 10
24 Answers to Chapter 11
25 Answers to Chapter 12
26 Answers to Chapter 13.

8.3. Review by: Steve Abbott.
The Mathematical Gazette 83 (496) (1999), 168-169.

The Texan mathematician R L Moore (1882-1974) pioneered a method of teaching mathematics which put the onus on the student to do the work. As Mary Ellen Rudin describes it,

He didn't lecture about mathematics at all. He put definitions on the board and gave us theorems to prove. Most of the time we didn't have the theorem proved that was supposed to be proved that day, and so we discussed whatever. We discussed life. And while we were doing that, he worked on us in various ways. ... [My thesis] was one of Moore's many unsolved problems. His technique was to feed all kinds of problems to us. He gave us lists of statements. Some were true, some were false, some he knew were true, some he knew were false, some were fairly easy to prove or disprove, others very hard.

lain Adamson's book follows the Moore method for the first half of the book, giving many definitions and stating many theorems and problems for the reader to prove or solve. The second half consists of solutions to the exercises appearing in the first part, rather compromising the spirit of the Moore method, though it is difficult to see an alternative. Adamson bases the course on the von Neumann - Bernays - Gödel axioms of set theory (NBG theory) rather than the more familiar Zermelo - Fraenkel (ZF) axioms. The first chapter begins uncompromisingly. ...
...
The success of this book as an example of the Moore method is somewhat dependent on the reader's willingness not to consult the answers until he or she has solved the problems. Also, a book cannot provide the encouragement that Mary Ellen Rudin referred to as Moore '[working] on us in various ways'. Nevertheless, I feel that there is much to be said for this style of book. Once a student has struggled with the definitions, exercises and theorems, a more standard text, with its 'scaffolding' will be that much easier to appreciate. The danger of the Moore method is that the student may well lose the plot if there is no-one to supply the overview.

The material covered in this book includes the NBG axioms, relations and functions, well-ordering, equivalent formulations of the Axiom of Choice, ordinals, cardinals and their arithmetic. It is not a book that a beginner can easily pick up in the middle, because there are so many terms and symbols defined on the way. However, there is a useful index.

Though it would be possible for a determined student to work through this book alone, I feel that it would more realistic to view it purely as a text-book which a lecturer could build a course around. The discussion of alternative methods of proof that fellow students had discovered would provide a vital element in the learning process.

8.4. Review by: Neil H Williams.
Studia Logica: An International Journal for Symbolic Logic 69 (3) (2001), 433-435.

This book is intended as a textbook for a first course in elementary axiomatic set theory, at the advanced undergraduate level. It provides a fairly standard development of set theory from the axioms, up to well orderings and simple cardinal and ordinal arithmetic. It stops well short of more advanced topics, like large cardinals, constructible sets, or forcing.

In the Preface, the author states that the book is written "based on the principle that for at least some branches of mathematics a good way for the student to learn is to be presented with a clear statement of the definitions of the terms with which the subject is concerned and then to be faced with a collection of problems involving the terms just defined". In following this principle, the book is in two parts, about 75 pages in each. Part One contains (with minimum discussion) the axioms and definitions, and 155 exercises based on them. There are no examples or theorems as such. All the development of the subject is left for the reader to do by working through the exercises. These exercises vary from simple examples and trivial propositions to significant theorems. Short hints are provided in the text for the majority of the exercises. Part Two contains full solutions to all the exercises. Heavy use of symbolism is avoided: the definitions and solutions are given for the most. part in standard "mathematical English".

The book opens with a one page summary of the underlying logic. (Too brief to be useful for a reader unfamiliar with symbolic logic, and moreover in his quest for brevity the author has mis-stated his definition of 'bound variable', and the 'substitute equals for equals' property of equality is incorrect as given.) Chapter 1 (First axioms of the theory NBG) presents the standard elementary set-building axioms. The axiomatic formulation is based on the Kelley-Morse system. Thus the variables in the system range over classes; a class that is a member of some class is a set. The axiom scheme of separation is stated in the form that for every formula $P(x)$ with free variable $x$ , there exists the class of all those sets $x$ which have the property $P(x)$. There is no restriction on the bound variables in $P(x)$: restricting all bound variables to range only over sets gives the weaker Von Neuman-Bernays-Gödel theory. (The strong form is given only for the simplicity of its statement. The weaker form suffices for all the results presented in the book).

Chapter 2 is on relations (as sets of ordered pairs). Chapter 3 is entitled "Functional relations and mappings". The author uses the phrase 'functional relation' in preference to the word 'function' for a relation which has the function property. However, a mapping is defined to be an ordered pair $((A, B), R)$ where $R$ is a functional relation with domain $A$ and range a subset of $B$. There is no motivation given for this definition. We need to look in the Preface, where we find that this book is a companion to the author's A General Topology Workbook (Birkhäuser, 1996). It would seem that 'mapping' is introduced in this way to ease the transition from topology to category theory.

The next chapter moves on to union, intersection and cartesian product of families of sets, including the statement of the axiom of choice in the form that the cartesian product of a family (indexed by a set) of non-empty sets is non-empty. Then come several chapters on equivalence relations, order relations, and well orderings. Next are chapters on ordinals, and on natural numbers. Then a few of the more frequently used equivalents of the axiom of choice: the well ordering principle and some maximal principles. The treatment ends with the definition of cardinals, and the beginnings of cardinal and ordinal arithmetic.

It seems to this reviewer that a major shortcoming of the book is the brevity of the discussion. There is no mention of any of the philosophical issues that are relevant to set theory. Apart from a couple of trite remarks in the Introduction there is no comment on the historical development of the subject. No attempt is made to put the definitions and results in their mathematical perspective. Rarely is there even any indication of whether an exercise is a routine example that once mastered should be put aside, or whether it is a significant result that sees frequent application. These shortcomings are perhaps not so serious if the book is used as an adjunct to a lecture course, where these matters could be dealt with in lectures. However a reader studying alone would need to consult other texts to gain motivation and maintain a sense of direction amongst the exercises.