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.