Elementary Number Theory and Algebra

Code: MA222 | L-T-P-C: 3-0-0-6

Number theory: Well ordering principle, principle of mathematical induction; Division algorithm, GCD and LCM, Euclidean algorithm, linear Diophantine equation; Primes, the fundamental theorem of arithmetic; Properties of congruences, linear congruences, chinese remainder theorem; Fermat's little theorem; Arithmetic functions, Mobius inversion formula, Euler's theorem; Primitive roots; Introduction to cryptography, RSA cryptosystem, distribution of primes.

Algebra: Groups, subgroups, cyclic groups, permutation groups, Cayley's theorem, cosets and Lagrange's theorem, normal subgroups, quotient groups, homomorphisms and isomorphism theorems; Rings, integral domains, ideals, quotient rings, prime and maximal ideals, ring homomorphisms, field of quotients, polynomial rings, factorization in polynomial rings, fields, characteristic of a field, field extensions, splitting fields, finite fields.

Texts:

1. D. M. Burton, Elementary Number Theory, 7th Ed., McGraw Hill, 2017.
2. J. A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1998.

References:

1. I. Niven, S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th Ed., Wiley-India, 1991.
2. G. A. Jones and J. M. Jones, Elementary Number Theory, Springer, 1998
3. K. H. Rosen, Elementary Number Theory and its Applications, Pearson, 2015
4. I. N. Herstein, Topics in Algebra, Wiley, 2004.
5. J. B. Fraleigh, A First Course in Abstract Algebra, Addison Wesley, 2002.