TCS Sem 1

Course: Abstract Algebra

Course code: 20XT23

Syllabus:

4  0  0  4

Groups: Introduction to algebraic structures: Groups – Definition and examples, properties of groups, Permutation groups, Symmetric groups, Cyclic groups, Check digit scheme.  (12)

Subgroups and Normal subgroups: Subgroups – Definition, Cosets and Lagrange’s theorem, Homomorphism, Isomorphism, Automorphism – Cayley’s theorem – Normal subgroups – Factor group – Fundamental theorem of group homomorphism. (12)

Group Coding: Coding of binary information and error detection – Group codes – Decoding and error correction. (8)

Rings: Definition and properties – Subrings, Ring of Quarternions, Integral domain – Homomorphism – Ideals and Quotient rings – Euclidean ring – Unique factorization theorem, Domain of Gaussian integers, Polynomial rings – Properties, Division algorithm – Factorization of polynomials – Primitive polynomials. (14)

Fields: Definition – Subfields – Finite fields – Structure of finite fields – GF(2^n ). (7)

Groups - Chapters 2,4,5

Subgroups and normal subgroups - Chapters 3,6,7,9,10

Group coding - Chapter 31 in Joseph A. Gallian, Contemporary Abstract Algebra, 8th edition, Brooks/Cole) and Chapter 16  in "Thomas W. Hungerford, Abstract Algebra an introduction, 3rd Edition, Brooks/Cole"

Rings - Chapters 12,13,15,14,18,16,17

Fields - Chapter 22

Geometric constructions - Chapter 23

Other Text Books:

1. Herstein, I.N., Topics in Algebra, John Wiley, 2012

2. Tremblay, J.P. and Manohar. R., Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw Hill, 2017

Reference Books:

1. Ron M. Roth, Introduction to Coding Theory, Cambridge University Press, 2016 

2. Ralph P. Grimaldi and Ramana B.V., Discrete and Combinatorial Mathematics: An Applied Introduction, Pearson Education, 2014

Web Sites:

https://isidore.co/calibre/get/pdf/4975

https://nptel.ac.in/courses/106/104/106104149/

http://www.d.umn.edu/~jgallian/

http://abstract.ups.edu/download/aata-20190710.pdf

Course Objectives:

CO1: To define and explain the concept of groups and  be able to apply it  in real life situations

CO2: To define and explain the concept of subgroups, cosets, homomorphism, isomorphism, automorphism, normal subgroups and factor groups and be apply to apply it in Lagrange’s theorem and fundamental theorem of homomorphism

CO3: To demonstrate coding and encoding of binary information with the use of group in coding theory. To define ring, subring, ring homomorphism, quotient ring, ideals and be able to relate them

CO4: To define polynomial rings and be able to apply its properties in division algorithm and factorization of polynomials

CO5: To describe finite fields and its structure and be able to apply it in geometric constructions