### Chapter Dependency Map

### 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.

#### Preliminaries

#### Complex numbers

#### Modular arithmetic

#### Modular arithmetic, decimals, and divisibility

#### Sets. Basic set properties

#### Functions

#### Introduction to cryptography

#### Sigma Notation

#### Polynomials

#### Symmetries

#### Permutations

#### Introduction to Groups

#### Further topics in cryptography

#### Equivalence relations and equivalence classes

#### Cosets and Factor Groups

#### Error Detecting and Correcting Codes

#### Isomorphisms of Groups

#### Homomorphisms of Groups

#### Group Actions

#### Introduction to Rings

#### Appendix

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.**

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**

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**

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

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

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

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)**

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)**

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)**

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)**

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)**

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**

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

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)**

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

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

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

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

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.

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.

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.