0
1854

# MA8551- ALGEBRA AND NUMBER THEORY Syllabus 2017 Regulation

MA8551- ALGEBRA AND NUMBER THEORY Syllabus 2017 Regulation

MA8551Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  ALGEBRA AND NUMBER THEORYÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â L T P C
Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  4 0 0 4
OBJECTIVES:

• 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 RINGSÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  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 POLYNOMIALSÂ  Â  Â  Â  Â  Â  Â  Â 12

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

## UNIT III DIVISIBILITY THEORY AND CANONICAL DECOMPOSITIONSÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 12

Division algorithm â€“ Base – b representations â€“ Number patterns â€“ Prime and composite numbers â€“ GCD â€“ Euclidean algorithm â€“ Fundamental theorem of arithmetic â€“ LCM.

## UNIT IV DIOPHANTINE EQUATIONS AND CONGRUENCESÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  12

Linear Diophantine equations â€“ Congruenceâ€˜s â€“ Linear Congruenceâ€˜s – Applications: Divisibility tests – Modular exponentiation-Chinese remainder theorem â€“ 2 x 2 linear systems.

## UNIT V CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONSÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â 12

Wilsonâ€˜s theorem â€“ Fermatâ€˜s little theorem â€“ Eulerâ€˜s theorem â€“ Eulerâ€˜s Phi functions â€“ Tau and Sigma functions.

Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  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.

TEXTBOOKS:

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

REFERENCES:

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