An Introduction to Number Theory

ABOUT THE COURSE:This course aims to introduce some basic topics in number theory. We will study quadratic number fields focusing on the structure of prime ideals and units in the ring of integers of the field. We will use binary quadratic forms to measure the failure of unique factorisation in the ring of integers of imaginary quadratic fields. Further, we will study classical Diophantine problems related to congruent numbers and rational cube sums and their relation with elliptic curves. Along the way, we will recall some basic concepts from abstract algebra which are required in this course.INTENDED AUDIENCE: BS/BSc/BMATH/MSc/MMATHPREREQUISITES:Familiarity with basic linear algebra will be useful.INDUSTRY SUPPORT: The topics in this course are relevant to Cryptography and Cybersecurity.

Syllabus

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