MA 419 (Basic Algebra)

Instructor : Sudhir R. Ghorpade, J K Verma

Course Description : The course is an introductory course to abstract algebra. It covers basics in group and ring theory, each covered by one professor. A variety of topics are covered:

Equivalence relations and partitions, Division algorithm for integers, primes, unique factorization, congruences, Chinese Remainder Theorem,

Euler j-function.

Permutations, sign of a permutation, inversions, cycles and transpositions.

Rudiments of rings and fields, elementary properties, polynomials in one and several variables, divisibility, irreducible polynomials, Division algorithm, Remainder Theorem, Factor Theorem, Rational Zeros Theorem, Relation between the roots and coefficients, Newton's Theorem on symmetric functions, Algebraic Numbers, etc

Groups, subgroups and factor groups, Lagrange's Theorem, homomorphisms, normal subgroups. Quotients of groups, Basic examples of groups (including symmetric groups, matrix groups, group of rigid motions of the plane and finite groups of motions).

Cyclic groups, generators and relations, Cayley's Theorem, group actions, Sylow Theorems.

The content can vary slightly with the instructor.

Logistics :

• Three hours of lecture and one tutorial per week
• Grading - 4-5 quizzes (best 3), midsem, endsem

Comments on Instructor : The two instructors teach well. Initial sections may appear easy but the later sections show a rapid increase in difficulty. It is a good idea not to miss any classes. Most questions are similar to or use the same ideas as the tutorial.

Who will find it interesting : Group and Ring theory is commonly used in a variety of areas like cryptography, symbolic computations, etc.