# Algebra III

Course TypeCourse CodeNo. Of Credits
Foundation CoreSUS1MA5094

Semester and Year Offered: Monsoon Semester 2013-14

Course Coordinator and Team: Kranti Kumar, Balchand Prajapati, Geetha Venkataraman

Email of course coordinator: balchand[at]aud[dot]ac[dot]in

Pre-requisites: Pre-requisite for this course is Mathematics at the XII grade level and 4 credits of SUS1MA506 (Algebra II)

Aim: This course helps students develop their ability to think abstractly within the setting offered through abstract and linear algebra. Such skills would add to a student’s ability to analyse and solve problems even in real life.

Course Outcomes:

After completing this course, students will be able

• to understand cyclic and abelian groups as well as direct products.
• familiar with important classes of rings, namely polynomial rings, Euclidean Domains, Principal Ideal Domains, Unique Factorisation Domains
• know Eigenvalues, Eigenvectors and diagonalization.

Brief description of modules/ Main modules:

The following topics will be covered in the course under the four main modules as described below.

Group Theory

Cayley’s theorem, automorphisms, external direct products: definition, examples and properties, criterion for external direct product of a finite number of finite cyclic groups to be cyclic, the group of units modulo n as an external direct product, applications of external direct product, fundamental theorem of finite abelian groups, the isomorphism classes of abelian groups, proof of the fundamental theorem.

Ring Theory

Polynomial rings: rings of polynomials over a commutative ring R, the division algorithm and its consequences; factorization of polynomials: definition of irreducible and reducible polynomials, reducibility tests, content of a polynomial, Gauss’s lemma, irreducibility tests including Eisenstein’s criterion, unique factorization in the polynomial ring over the integers, weird dice: an application of unique factorization; divisibility in integral domains: associates, irreducible and primes, unique factorization domains, principal ideal domains, Euclidean domains, relationship between Euclidean domains, principal ideal domains and unique factorization domains.

Linear Algebra

Change of basis, applications to difference equations, applications to Markov chains, eigenvalues and eigenvectors, eigenspaces and similarity, representation by a diagonal matrix.

Assessment Details with weights:

 S.No Assessment Date/period in which Assessment will take place Weightage 1 Class test End August 10% 2 Mid Semester Exam End September/ early October 25% 3 Lab/ Home Assignments Throughout the semester 15% 4 Presentation/ Viva End October/ early November 15% 5 End Semester Exam As per AUD Academic Calendar 35%