ABOUT THE COURSE:Number theory is a study of Diophantine equations, in other words, polynomial equations with integer or rational coefficients for which we seek integer or rational solutions. For example, Pythagorean triplets x^2+y^2 = z^2. More generally, Fermat equation x^n+y^n = z^n. Another famous example is the Catalan equation x^n+1 = y^m etc. Although we look for integer solutions, it is often useful to extend our number systems and work in these extended number systems. For instance, x^2+y^2 factorises as (x+iy)(x-iy), where i is a square root of -1. Hence it makes more sense to work over Gaussian numbers Q(i). More generally, we can consider any polynomial f(x) with integer coefficients and obtain a finite extension of rational numbers by attaching roots of f(x) = 0. These are called number fields. In this Algebraic Number Theory course we study such number fields, and various objects attached to it. Some of these objects and invariants are Ring of integers Units in the ring of integersDiscriminantDifferentIdeal class groupL-functions.We will define these objects, study their properties, study algorithms to compute them, and prove several properties about them.INTENDED AUDIENCE: Masters and PhD students.PREREQUISITES: Algebra, Galois theory.INDUSTRY SUPPORT: Cryptography based.
Overview
Syllabus
Week 1: Study of number fields, definition of the ring of integers. Definition of norm and trace.
Week 2:Definition of absolute and relative discriminant. Computation of discriminant. Computation of the ring of integers.
Week 3:Definition and properties of Dedekind domains. Proof that the ring of integers is a Dedekind domain. Factorisation of extension of prime ideals in a finite extension of number fields.
Week 4:Embeddings of a number field in complex numbers. A result from geometry of numbers. Finiteness of class groups.
Week 5:Computation of class groups, including several examples. Applications to Diophantine equations of computations of class groups.
Week 6:Dirichlet’s unit theorem.
Week 7:Extension and norm of ideals in field extensions. Maps between class groups of extensions. Decomposition subgroups, inertia subgroups, Frobenius elements etc. Localisation, residue field.
Week 8:Valuations in a number fields. Local fields. Hensel’s lemma and applications.
Week 9:Field extensions of local fields, ramification, different, inertia subgroups etc.
Week 10:Study of special number fields. Imaginary quadratic fields, real quadratic fields, cubic fields, cyclotomic fields.
Week 11:Definition of ray class field as a generalisation of ideal class group. Some statements from class field theory without proofs.
Week 12:Definition of zeta functions and L-functions. Statements of their analytic properties without proofs. Dirichlet Class number formula.
Week 2:Definition of absolute and relative discriminant. Computation of discriminant. Computation of the ring of integers.
Week 3:Definition and properties of Dedekind domains. Proof that the ring of integers is a Dedekind domain. Factorisation of extension of prime ideals in a finite extension of number fields.
Week 4:Embeddings of a number field in complex numbers. A result from geometry of numbers. Finiteness of class groups.
Week 5:Computation of class groups, including several examples. Applications to Diophantine equations of computations of class groups.
Week 6:Dirichlet’s unit theorem.
Week 7:Extension and norm of ideals in field extensions. Maps between class groups of extensions. Decomposition subgroups, inertia subgroups, Frobenius elements etc. Localisation, residue field.
Week 8:Valuations in a number fields. Local fields. Hensel’s lemma and applications.
Week 9:Field extensions of local fields, ramification, different, inertia subgroups etc.
Week 10:Study of special number fields. Imaginary quadratic fields, real quadratic fields, cubic fields, cyclotomic fields.
Week 11:Definition of ray class field as a generalisation of ideal class group. Some statements from class field theory without proofs.
Week 12:Definition of zeta functions and L-functions. Statements of their analytic properties without proofs. Dirichlet Class number formula.
Taught by
Prof. Mahesh Kakde
