Groups : Definition - Properties - Homomorphism - Isomorphism - Cyclic groups - Cosets - Lagrange's theorem. Rings: Definition - Sub rings - Integral domain - Field - Integer modulo n - Ring homomorphism.
Rings - Polynomial rings - Irreducible polynomials over finite fields - Factorization of polynomials over finite fields.
Division algorithm – Base – b representations – Number patterns – Prime and composite numbers – GCD – Euclidean algorithm – Fundamental theorem of arithmetic – LCM
Linear Diophantine equations – Congruence‘s – Linear Congruence‘s - Applications: Divisibility tests - Modular exponentiation-Chinese remainder theorem – 2 x 2 linear systems.
Wilson‘s theorem – Fermat‘s little theorem – Euler‘s theorem – Euler‘s Phi functions – Tau and Sigma functions.
Reference Book:
● Lidl, R. and Pilz, G, "Applied Abstract Algebra", Springer Verlag, New Delhi, 2nd Edition, 2013. ● David Joyce,”Introduction to Modern Algebra”2017 ● San Ling and Chaoping Xing, ―Coding Theory – A first Course‖, Cambridge Publications, Cambridge, 2004. ● Niven, I., Zuckerman.H.S., and Montgomery, H.L., ―An Introduction to Theory of Numbers‖, John Wiley and Sons , Singapore, 2004. ● Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 2002.
Text Book:
● Grimaldi, R.P and Ramana, B.V., "Discrete and Combinatorial Mathematics", Pearson Education, 5th Edition, New Delhi, 2007. ● Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 7TH Edition 2010. ● Martyn R. Dixon ,Leonid A. Kurdachenko , Igor Ya. Subbotin ,”Algebra and Number Theory ; An Integrated Approach”, John Wiley & Sons,2010.
