Groups: Definition - Properties - Homomorphism - Isomorphism - Cyclic groups - Cosets - Lagrange's theorem. Rings: Definition - Sub rings - Integral domain - Field - Integer modulo n - Ring homomorphism.
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:
1. Lidl, R. and Pilz, G, "Applied Abstract Algebra", Springer Verlag, New Delhi, 2nd Edition, 2013 2. David Joyce,â€Introduction to Modern Algebraâ€2017 3. San Ling and Chaoping Xing, ―Coding Theory – A first Course‖, Cambridge Publications, Cambridge, 2004. 4. Niven, I., Zuckerman.H.S., and Montgomery, H.L., ―An Introduction to Theory of Numbers‖, John Wiley and Sons , Singapore, 2004. 5. Koshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 2002.
Text Book:
1. Grimaldi, R.P and Ramana, B.V., "Discrete and Combinatorial Mathematics", Pearson Education, 5th Edition, New Delhi, 2007. 2.cKoshy, T., ―Elementary Number Theory with Applications‖, Elsevier Publications, New Delhi, 7TH Edition 2010. 3. Martyn R. Dixon,Leonid A. Kurdachenko,Igor Ya. Subbotin,â€Algebra and Number Theory ; An Integrated Approachâ€, John Wiley & Sons,2010
