UM 203: Elementary Algebra and Number Theory

Credits: 3:0

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 :

  1. Burton, D. M., Elementary Number Theory ,McGraw Hill.
  2. Niven, Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edition ,Wiley Student Editions.
  3. Fraleigh, G., A First Course in Abstract Algebra, 7th edition ,Pearson.

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Contact: +91 (80) 2293 2711, +91 (80) 2293 2625
E-mail: chairman.math[at]iisc[dot]ac[dot]in