Week 1: Abelian groups, subgroups and quotient groups, finite abelian groups and finitely generated abelian groups
Week 2: Commutative Rings, ideals, fields, Polynomial rings, zero sets of ideals in polynomial rings.
Week 3: The groups Z/nZ and (Z/nZ)*, Euler’s theorem and Wilson Theorem. Chinese Remainder Theorem,
Week 4: Classification of finite fields, the multiplicative subgroup of a finite field,
Week 5: Law of quadratic reciprocity, Quadratic fields, Containment of quadratic fields inside cyclotomic fields
Week 6: UFD and PID, Ring of integers of quadratic fields, Various examples.
Week 7: Binary quadratic forms
Week 8: Ideal class groups of imaginary quadratic fields
Week 9: Units in the ring of integers of quadratic fields, Diophantine problem: Bramhagupta-Pell’s equation
Week 10: Infinite descent, n=4 case of the Fermat’s Last Theorem, Diophantine problem of congruent numbers, relation with elliptic curves
Week 11: Diophantine problem on rational cube sums, relation with elliptic curves
Week 12: Group law of elliptic curves, statements of Nagell-Lutz Theorem and Mordell-Weil Theorem
DOWNLOAD APP
FOLLOW US