Syllabus
|
Course Code: Elective 4 MMATH21 -406 Course Name: Algebraic Number Theory |
||
| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
|---|---|---|
| 1 | Norm and trace of algebraic numbers and algebraic integers, Bilinear map on algebraic number field K. Integral basis and discriminant of an algebraic number field, Index of an element of K, Ring Ok of algebraic integers of an algebraic number field K. Ideals in the ring of algebraic number field K. | |
| 2 | Integrally closed domains. Dedekind domains. Fractional ideals of K. Factorization of ideals as a product of prime ideals in the ring of algebraic integers of an algebraic number field K. G.C.D. and L.C.M. of ideals in Ok. Chinese Remainder theorem, order of ideal in prime ideal, ramification degree of prime ideals, different of an algebraic number field K, Dedekind theorem. | |
| 3 | Euclidean rings. Hurwitz Lemma and Hurwitz constant. Equivalent fractional ideals. Ideal class group. Finiteness of the ideal class group. Class number of the algebraic number field K. Diophantine equations, Minkowski’s bound. | |
| 4 | Legendre Symbol, Jacobi symbol, Gauss sums, Law of quadratic reciprocity, Quadratic fields, Primes in special progression, class number of quadratic fields. | |
