Chapter Dependency Map

Chapter Dependency Image

Chapter Contents

The following are the chapter's for the book with hyperlinks to the respective page on the online textbook. A summary of each chapter is also provided.

  1. Preliminaries

  2. A review of properties of integers, rationals, and reals, at the high school level. We only review the properties; we do not formally construct these number systems. Some remedial exercises are included. Used in: All other chapters.

  3. Complex numbers

  4. Basic properties of complex arithmetic, polar form, exponentiation and roots. Some exercises require proofs of complex number properties. The last section presents applications to signal processing and fractals. Used in: Symmetries (10); all theory chapters

  5. Modular arithmetic

  6. ℤnℤn is the gold-standard example for finite groups and rings. Arithmetic properties, Euclidean algorithm, Diophantine equations; We bring out homomorphism properties (without the terminology). Used in: all subsequent chapters

  7. Modular arithmetic, decimals, and divisibility

  8. Application of modular arithmetic to decimal representation of real numbers (in arbitrary bases) and divisibility rules.

  9. Sets. Basic set properties

  10. Can be skipped if students have an adequate background in discrete math. Used in: functions

  11. Functions

  12. Basic ideas of domain, range, into, onto, bijection. This chapter can be skipped if students have an adequate background. Used in: all subsequent chapters

  13. Introduction to cryptography

  14. Explains the concepts of public and private key cryptography, and describes some classic cyphers as well as RSA. Used in: Further topics in Cryptography (13)

  15. Sigma Notation

  16. This chapter prepares for the "polynomials" chapter. Sigma notation is useful in linear algebra as well. Can be skipped if students are already familiar with this notation. Used in: Polynomials (9)

  17. Polynomials

  18. Fundamental example of rings. Euclidean algorithm for polynomials over fields. FTOA, prove easy part and discuss the hard part. Will cover this again more rigorously in later chapter. Used in: Introduction to Groups (12), Introduction to Rings (20)

  19. Symmetries

  20. Symmetries are a special case of permutations. They are treated first because they are easily visualizable, and because they connect algebraic aspects to geometry as well as complex numbers. Used in: Permutations (11)

  21. Permutations

  22. In light of Cayley's theorem, this example is key to the understanding of finite groups. Students are introduced to the mechanics of working with permutations, including cycle multiplication. Cycle structure is explored, as are even and odd permutations. Used in: Introduction to Groups (12)

  23. Introduction to Groups

  24. This chapter introduced basic properties of groups, subgroups, and cyclic groups, drawing heavily on the examples presented in previous chapters. Used in: all subsequent chapters

  25. Further topics in cryptography

  26. Diffie-Hellman key exchange, elliptic curve cryptography over ℝ and over ℤp

  27. Equivalence relations and equivalence classes

  28. This material is necessary for understanding cosets. This chapter may be skipped if students have seen them before. Used in: Cosets and Factor Groups (15)

  29. Cosets and Factor Groups

  30. Introductory properties, Lagrange's theorem, Fermat's Theorem, simple groups. Used in: all subsequent chapters.

  31. Error Detecting and Correcting Codes

  32. A discussion of block codes. Some knowledge of linear algebra is required.

  33. Isomorphisms of Groups

  34. Examples and basic properties; direct products (internal and external); classification of abelian groups up to isomorphism. Used in: all subsequent chapters

  35. Homomorphisms of Groups

  36. Kernel of homomorphism; properties; first isomorphism theorem. Used in: all subsequent chapters

  37. Group Actions

  38. Besides basic definitions, this chapter contains a long discussion of group actions applied to regular polyhedral, as well as the universal covering space of the torus.

  39. Introduction to Rings

  40. Includes definitions and examples; subrings and product rings; extending polynomial rings to fields; isomorphisms and homomorphisms; ideals; principal ideal domains; prime ideals and unique factorization domains; division rings; fields; algebraic extensions.

  41. Appendix

  42. Induction Proofs-patterns and examples. Some proofs in the book require induction. This section gives the background needed for students to write formal induction proofs.