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SYLLABUS

Unit - 1

1. 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 - 2

2. Finite Fields and Polynomials Rings - Polynomial Rings - Irreducible Polynomials Over Finite Fields - Factorization of Polynomials Over Finite Fields.

Unit - 3

3. 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 - 4

4. Diophantine Equations and Congruences Linear Diophantine Equations - Congruences - Linear Congruences - Applications: Divisibility Tests - Modular Exponentiation - Chinese Remainder Theorem - 2 × 2 Linear Systems.

Unit - 5

5. Classical Theorems and Multiplicative Functions Wilson’s Theorem - Fermat’s Little Theorem - Euler’s Theorem - Euler’s Phi Functions - Tau and Sigma Functions.