BE-CSE AND BE-IT ALGEBRA AND NUMBER THEORY SYLLABUS

 

ANNA UNIVERSITY, CHENNAI
AFFILIATED INSTITUTIONS

MA8551

ALGEBRA AND NUMBER THEORY

B.E-CSE AND B.E -IT

REGULATION-2017 

III-YEAR ,V-SEMESTER   

COURSE OBJECTIVE:

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

COURSE OUTCOME
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.

 

UNIT-I     GROUPS AND RINGS                                                             (12)

Groups : Definition - Properties - Homomorphism - Isomorphism - Cyclic groups - Co sets - 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 DECOMPOSITION(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 CONGRUENCE'S         (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.

 

 

TEXTBOOKS:

 

   * Grimaldi, R.P and Ramana, B.V., "Discrete and Combinatorial Mathematics", Pearson  Education, $5^th$ Edition, New Delhi, 2007
   *  Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 2002. 

 

 

REFERENCES :

 

     * Lidl, R. and Pitz, G, "Applied Abstract Algebra", Springer Verlag, New Delhi, $2^nd$ 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.