Divisibility and Euclid’s algorithm; Fundamental theorem of arithmetic; Infinitude of primes;
Congruences; (Reduced) residue systems, Application to sums of squares; Chinese Remainder Theorem;
Solutions of polynomial congruences, Hensel’s lemma; A few arithmetic functions (in particular, discussion
of the floor function); the Mobius inversion formula; Recurrence relations; Basic combinatorial number
theory (pigeonhole principle, inclusion-exclusion, etc.); Primitive roots and power residues, Quadratic
residues and the quadratic reciprocity law, the Jacobi symbol; Some Diophantine equations, Pythagorean
triples, Fermat’s descent, examples; Definitions of groups, rings and fields, motivations, examples and basic
properties; polynomial rings over fields, factorisation of polynomials, content of a polynomial and Gauss’
lemma, Eisenstein’s irreducibility criterion; Elementary symmetric polynomials, the fundamental theorem
on Symmetric polynomials; Algebraic and transcendental numbers (an introduction).
Suggested books and references:
Burton, D. M., Elementary Number Theory, McGraw Hill.
Niven, Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edition, Wiley Student Editions.
Fraleigh, G., A First Course in Abstract Algebra, 7th edition, Pearson.