ANNA UNIVERSITY ALGEBRA AND NUMBER THEORY MA8551 SYLLABUS REGULATION 2017

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 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 x 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. 

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.