Congruences, Polynomials, and Group Theory

by WWL Chen

This set of notes has been compiled over the last few years. All 4 chapters have been in use at Macquarie University since 1999, and cover material in first year mathematics that does not fall neatly into either calculus or linear algebra.

The following is the suggested order for the presentation of the material:

MATH135:

• Chapters 1 and 2.

MATH132:

• Chapter 2.

MATH136:

• Chapter 3.

MATH133:

• Chapters 3 and 4.

To read the notes, click the chapters below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

Chapter 1: REVIEW ON TRIGONOMETRY (written with XT Duong)

• Radian and Arc Length
• The Trigonometric Functions
• Some Trigonometric Identities

Chapter 2: POLYNOMIALS

• Introduction
• Polynomial Equations
• Division of Polynimials
• Roots of Polynomials
• Fundamental Theorem of Algebra
• Roots of Real Polynomials
• Rational Functions
• Greatest Common Divisor

Chapter 3: CONGRUENCES

• Introduction
• Arithmetic of Congruences
• Linear Congruences
• Special Divisibility Rules
• Further Discussion

Chapter 4: INTRODUCTION TO GROUP THEORY

• Some Examples
• Formal Definition
• Subgroups
• Order
• Cyclic Groups
• Further Discussion
• Groups of Small Order