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Publisher:

John Wiley

Publication Date:

2005

Number of Pages:

354

Format:

Hardcover

Edition:

5

Price:

105.95

ISBN:

0-471-43335-7

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Frederick M. Butler

05/28/2005

The release of this fifth edition of Durbin's *Modern Algebra* has given me an opportunity to peruse a classic modern algebra book which I never had an opportunity to use in class. A colleague of mine, who has been at this longer than I have, said he has not found an undergraduate algebra text better than this one. Looking through the book, it is easy to see why he has such a high opinion.

There is a very nice introduction, which tries to show students the general nature of the book's main ideas, and how they developed over time. I particularly liked the section on the three classic (impossible) straightedge and compass constructions. The author uses a picture and an equation to make the constructions concrete to the reader. For example, the squaring the circle construction is represented by pictures of a circle of radius r and a square of side s, with the equation πr^{2} = s^{2} below. As an undergraduate I always found these constructions confusing, but the explanations given by Durbin are crystal clear. There are also several appendices, on sets, proofs, induction, and linear algebra, which are well done.

The book contains chapters on the standard topics one has come to expect in a modern algebra textbook. But none of these chapters are overwhelming, each averaging around twenty pages. The book does not try to give exhaustive coverage; in fact, one of its greatest strengths is its brevity. Durbin's approach is to be as clear and concise as possible, while rigorously developing the subject. He gives just what is needed to understand the topic, build on the previous chapters, and develop the theory. The proofs are also very clear, written in such a way as to be accessible to an undergraduate student.

There are also chapters on a few less standard subjects, including symmetry, cryptography and algebraic coding, and lattices and Boolean algebras. An instructor lacking the time to cover these topics could easily assign some sort of project based on these chapters. The presentation is clear enough that a student could learn the material mostly independently. The changes from the past edition aren't too extensive, consisting mainly of an expanded chapter on field and Galois theory, a new chapter on cryptography, and some additional problems.

My one complaint is that the exercises could be a bit more challenging. Although this is one of the changes addressed in this new edition, the exercises still consist mostly of verifying or constructing fairly straightforward examples. For undergraduates first learning the subject, certainly some routine exercises are needed. But it seems to me the book could benefit from a few challenge problems in each problem section. This is a minor point, however, in a book with very strong exposition that would make an outstanding text for an undergraduate abstract algebra course.

Frederick M. Butler is Assistant Professor of Mathematics at the Institute for Mathematics Learning of West Virginia University.

ntroduction.

I. Mappings and Operations.

1 Mappings.

2 Composition. Invertible Mappings.

3 Operations.

4 Composition as an Operation.

II. Introduction to Groups.

5 Definition and Examples.

6 Permutations.

7 Subgroups.

8 Groups and Symmetry.

III. Equivalence. Congruence. Divisibility.

9 Equivalence Relations.

10 Congruence. The Division Algorithm.

11 Integers Modulo n.

12 Greatest Common Divisors. The Euclidean Algorithm.

13 Factorization. Euler's Phi-Function.

IV. Groups.

14 Elementary Properties.

15 Generators. Direct Products.

16 Cosets.

17 Lagrange's Theorem.

18 Isomorphism.

19 More on Isomorphism.

20 Cayley's Theorem.

V. Group Homomorphisms.

21 Homomorphisms of Groups. Kernels.

22 Quotient Groups.

23 The Fundamental Homomorphism Theorem.

VI. Introduction to Rings.

24 Definition and Examples.

25 Integral Domains. Subrings.

26 Fields.

27 Isomorphism. Characteristic.

VII. The Familiar Number Systems.

28 Ordered Integral Domains.

29 The Integers.

30 Field of Quotients. The Field of Rational Number.

31 Ordered Fields. The Field of Real Numbers.

32 The Field of Complex Numbers.

33 Complex Roots of Unity.

VIII. Polynomials.

34 Definition and Elementary Properties.

35 The Division Algorithm.

36 Factorization of Polynomials.

37 Unique Factorization Domains.

IX. Quotient Rings.

38 Homomorphisms of Rings. Ideals.

39 Quotient Rings.

40 Quotient Rings of F[X].

41 Factorization and Ideals.

X. Field Extensions.

42 Simple Extensions.

43 Degrees of Extensions.

44 Splitting Fields.

45 Finite Fields.

XI. Galois Theory.

46 Galois Groups.

47 Separability and Normality.

48 Fundamental Theorem of Galois Theory.

49 Solvability by Radicals.

XII. Geometric Constructions.

50 Three Famous Problems.

51 Constructible Numbers.

52 Impossible Constructions.

XIII. Applications of Permutation Groups.

53 Groups Acting on Sets.

54 Burnside's Counting Theorem.

55 Sylow's Theorem.

XIV. Symmetry.

56 Finite Symmetry Groups.

57 Infinite Two-dimensional Symmetry Groups.

58 On Crystallographic Groups.

59 The Euclidean Group.

XV. Cryptography and Algebraic Coding.

60 RSA Algorithm.

61 Introduction to Algebraic Coding.

62 Linear Codes.

63 Standard Decoding.

64 Error Probability.

XVI. Lattices and Boolean Algebras.

65 Partially Ordered Sets.

66 Lattices.

67 Boolean Algebras.

68 Finite Boolean Algebras.

69 Switching.

A. Sets.

B. Proofs.

C. Mathematical Induction.

D. Linear Algebra.

E. Solutions to Selected Problems.

Photo Credit List.

Index of Notation.

Index.

I. Mappings and Operations.

1 Mappings.

2 Composition. Invertible Mappings.

3 Operations.

4 Composition as an Operation.

II. Introduction to Groups.

5 Definition and Examples.

6 Permutations.

7 Subgroups.

8 Groups and Symmetry.

III. Equivalence. Congruence. Divisibility.

9 Equivalence Relations.

10 Congruence. The Division Algorithm.

11 Integers Modulo n.

12 Greatest Common Divisors. The Euclidean Algorithm.

13 Factorization. Euler's Phi-Function.

IV. Groups.

14 Elementary Properties.

15 Generators. Direct Products.

16 Cosets.

17 Lagrange's Theorem.

18 Isomorphism.

19 More on Isomorphism.

20 Cayley's Theorem.

V. Group Homomorphisms.

21 Homomorphisms of Groups. Kernels.

22 Quotient Groups.

23 The Fundamental Homomorphism Theorem.

VI. Introduction to Rings.

24 Definition and Examples.

25 Integral Domains. Subrings.

26 Fields.

27 Isomorphism. Characteristic.

VII. The Familiar Number Systems.

28 Ordered Integral Domains.

29 The Integers.

30 Field of Quotients. The Field of Rational Number.

31 Ordered Fields. The Field of Real Numbers.

32 The Field of Complex Numbers.

33 Complex Roots of Unity.

VIII. Polynomials.

34 Definition and Elementary Properties.

35 The Division Algorithm.

36 Factorization of Polynomials.

37 Unique Factorization Domains.

IX. Quotient Rings.

38 Homomorphisms of Rings. Ideals.

39 Quotient Rings.

40 Quotient Rings of F[X].

41 Factorization and Ideals.

X. Field Extensions.

42 Simple Extensions.

43 Degrees of Extensions.

44 Splitting Fields.

45 Finite Fields.

XI. Galois Theory.

46 Galois Groups.

47 Separability and Normality.

48 Fundamental Theorem of Galois Theory.

49 Solvability by Radicals.

XII. Geometric Constructions.

50 Three Famous Problems.

51 Constructible Numbers.

52 Impossible Constructions.

XIII. Applications of Permutation Groups.

53 Groups Acting on Sets.

54 Burnside's Counting Theorem.

55 Sylow's Theorem.

XIV. Symmetry.

56 Finite Symmetry Groups.

57 Infinite Two-dimensional Symmetry Groups.

58 On Crystallographic Groups.

59 The Euclidean Group.

XV. Cryptography and Algebraic Coding.

60 RSA Algorithm.

61 Introduction to Algebraic Coding.

62 Linear Codes.

63 Standard Decoding.

64 Error Probability.

XVI. Lattices and Boolean Algebras.

65 Partially Ordered Sets.

66 Lattices.

67 Boolean Algebras.

68 Finite Boolean Algebras.

69 Switching.

A. Sets.

B. Proofs.

C. Mathematical Induction.

D. Linear Algebra.

E. Solutions to Selected Problems.

Photo Credit List.

Index of Notation.

Index.

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