OMA353 Algebra and Number Theory Syllabus:

OMA353 Algebra and Number Theory Syllabus – Anna University Regulation 2021

COURSE OBJECTIVES :

 To introduce the basic notions of groups, rings, fields which will then be used to solve related problems.
 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

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

Rings – Polynomial rings – Irreducible polynomials over finite fields – Factorization of polynomials over finite fields.

UNIT III DIVISIBILITY THEORY AND CANONICAL DECOMPOSITIONS

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

Linear Diophantine equations – Congruence’s – Linear Congruence’s – Applications : Divisibility tests – Modular exponentiation – Chinese remainder theorem – 2×2 linear systems.

UNIT V CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONS

Wilson’s theorem – Fermat’s Little theorem – Euler’s theorem – Euler’s Phi functions – Tau and Sigma functions.

TOTAL: 45 PERIODS
COURSE OUTCOMES:

CO1 Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts.
CO2 Demonstrate accurate and efficient use of advanced algebraic techniques.
CO3 The students should be able to demonstrate their mastery by solving non-trivial problems related to the concepts, and by proving simple theorems about the, statements proven by the text

TEXT BOOKS :

1. Grimaldi, R.P and Ramana, B.V., “Discrete and Combinatorial Mathematics”, Pearson Education, 5th Edition, New Delhi, 2007.
2. Thomas Koshy, “Elementary Number Theory with Applications”, Elsevier Publications , New Delhi , 2002.

REFERENCES:

1. San Ling and Chaoping Xing, “Coding Theory – A first Course”, Cambridge Publications, Cambridge, 2004.
2. Niven.I, Zuckerman.H.S., and Montgomery, H.L., “An Introduction to Theory of Numbers” , John Wiley and Sons , Singapore, 2004.
3. Lidl.R., and Pitz. G, “Applied Abstract Algebra”, Springer Verlag, New Delhi, 2nd Edition ,2006.