Unit 1

Number Theory: Mathematical induction, Division Algorithm, Greatest Common Divisor, Euclidean Algorithm for finding GCD, Linear Diophantine Equation, Primes Numbers and Fundamental Theorem of Arithmetic, Euclid’s proof of Infinitude of Primes.Basic properties of Congruences, Linear Congruences and Chinese Remainder Theorem, Fermat’s Little theorem, Wilson’s Theorem, Euler ϕ Function and its Properties, Euler’s Theorem.

Unit 2

Definition and examples of Groups, some elementary properties of groups, Order of a Group, Subgroups, Cyclic Groups, Classification of Subgroups of Cyclic Groups,Permutation Groups, Cycle Notation, Properties of Permutations, Isomorphism of Groups.

Unit 3

Left and Right Cosets, Properties of Cosets, Lagrange’s Theorem and consequences, Normal Subgroups and Factor / Quotient Groups, Group Homomorphisms, Kernel.

Unit 4

Rings, Properties of Rings, Subrings, Integral Domains, Fields, Characteristic of a Ring.

Unit 5

Ideals and Factor / Quotient Rings, Prime Ideals and Maximal Ideals, Ring Homomorphisms and Field of Quotients.

Course Outcome:
CO1: Ability to demonstrate insight into abstract algebra with focus on axiomatic theories CO2: Ability to apply algebraic ways of thinking
CO3: Ability to demonstrate knowledge and understanding of fundamental concepts including groups, subgroups, normal subgroups, homomorphisms and isomorphism
CO4: Ability to demonstrate knowledge and understanding of rings, fields and their properties
CO5: Ability to prove fundamental results and solve algebraic problems using appropriate techniques

Textbook:
David M. Burton, Elementary Number Theory, 7th edition, McGraw-Hill. Joseph A. Gallian, Contemporary Abstract Algebra, 4th edition, Narosa.

References:
John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson.
I.N. Herstein, Topics in Algebra, 2nd edition, John Wiley.