To introduce the basic notions of groups, rings, fields which will then be used to solve related problems.
To introduce and apply the concepts of rings, finite fields and polynomials.
To understand the basic concepts in number theory
To examine the key questions in the Theory of Numbers.
To give an integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.
UNIT I GROUPS AND RINGS 12
Groups : Definition – Properties – Homomorphism – Isomorphism – Cyclic groups – Cosets – Lagrange’s theorem. Rings: Definition – Sub rings – Integral domain – Field – Integer modulo n – Ring homomorphism.
UNIT II FINITE FIELDS AND POLYNOMIALS 12
Rings – Polynomial rings – Irreducible polynomials over finite fields – Factorization of polynomials over finite fields.
UNIT III DIVISIBILITY THEORY AND CANONICAL DECOMPOSITIONS 12
Division algorithm – Base – b representations – Number patterns – Prime and composite numbers – GCD – Euclidean algorithm – Fundamental theorem of arithmetic – LCM.
UNIT IV DIOPHANTINE EQUATIONS AND CONGRUENCES 12
Linear Diophantine equations – Congruence‘s – Linear Congruence‘s – Applications: Divisibility tests – Modular exponentiation-Chinese remainder theorem – 2 x 2 linear systems.
UNIT V CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONS 12
Wilson‘s theorem – Fermat‘s little theorem – Euler‘s theorem – Euler‘s Phi functions – Tau and Sigma functions.
TOTAL: 60 PERIODS OUTCOMES: Upon successful completion of the course, students should be able to:
Apply the basic notions of groups, rings, fields which will then be used to solve related problems.
Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts.
Demonstrate accurate and efficient use of advanced algebraic techniques.
Demonstrate their mastery by solving non – trivial problems related to the concepts, and by proving simple theorems about the, statements proven by the text.
Apply integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.
- Grimaldi, R.P and Ramana, B.V., “Discrete and Combinatorial Mathematics”, Pearson Education, 5th Edition, New Delhi, 2007.
- Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 2002.
- Lidl, R. and Pitz, G, “Applied Abstract Algebra”, Springer Verlag, New Delhi, 2nd Edition, 2006.
- Niven, I., Zuckerman.H.S., and Montgomery, H.L., ―An Introduction to Theory of Numbers‖, John Wiley and Sons , Singapore, 2004.
- San Ling and Chaoping Xing, ―Coding Theory – A first Course‖, Cambridge Publications, Cambridge, 2004.