Algebra

This book is based on a course Algebra. We write this book as per the revised syllabus of F.Y. B.Sc. Mathematics, revised by Savitribai Phule Pune University, Pune, implemented from June 2019. Algebra lies at the very foundation of diverse areas like coding theory, cryptography and many branches of theoretical computer science.

In chapter 1, we study set, relations and functions which builds strong foundation in the study of mathematics. Firstly, we deals with basic notions of sets and standard operations on sets. Then we give a brief introduction to various types of relations. We study functions which are building block in mathematics. We have also discussed image and inverse image of subsets under functions which is very useful in higher studies in mathematics.

In the second chapter, we discuss induction principle and well ordering principle on which many proofs in mathematics depend upon. We then study: integers and divisibility properties of integers, division algorithm and its applications, greatest common divisor and least common multiple and their properties, Euclidean algorithm and its applications.

The third chapter deals primes and theory of congruences. We study prime numbers and its properties. Furthermore, we study theory of congruences by means of several properties. We study a celebrated theorem called Fermat's theorem and its applications.

In the fourth chapter, we study the properties of complex numbers. Complex number are useful in electronics, uid dynamics and many other disciplines.

1. Sets Relations and Functions

2. Divisibility Theory in the Integers

3. Primes and the theory of Congruence

4. Complex Numbers