OMA353 Syllabus - Algebra And Number Theory - 2021 Regulation - Open Elective | Anna University
OMA353 Syllabus - Algebra And Number Theory - 2021 Regulation - Open Elective | Anna University
OMA353 |
ALGEBRA AND NUMBER THEORY |
L T P C |
---|
3003
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.
• 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 |
9 |
---|
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 |
9 |
---|
Rings - Polynomial rings - Irreducible polynomials over finite fields - Factorization of polynomials over finite fields.
UNIT III |
DIVISIBILITY THEORY AND CANONICAL DECOMPOSITIONS |
9 |
---|
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 |
9 |
---|
Linear Diophantine equations – Congruence’s – Linear Congruence’s - Applications : Divisibility tests - Modular exponentiation - Chinese remainder theorem – 2x2 linear systems.
UNIT V |
CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONS |
9 |
---|
Wilson’s theorem – Fermat’s Little theorem – Euler’s theorem – Euler’s Phi functions – Tau and Sigma functions.
TOTAL: 45 PERIODS
OUTCOMES:
• 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.
• 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
• Demonstrate accurate and efficient use of advanced algebraic techniques.
• 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.
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.
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.
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