Math 686R: Topics in Algebraic Number Theory.

Catalog Information

Title

Topics in Algebraic Number Theory.

Credit Hours

3

Prerequisite

Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

Description

Current topics of research interest.

Desired Learning Outcomes

To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.

Prerequisites

Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

Minimal learning outcomes

These cannot be specified uniquely for a topics course. The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility.

1. Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions:
   • Number fields
   • Algebraic integers in a number field
   • Integral bases
   • Discriminant
   • Norms of ideals
   • Finiteness of ideals of bounded norm
   • Class number
   • Finiteness of class number
   • Dedekind's Unique Factorization Theorem for ideals of a number field
2. Geometry of numbers:
   • Minkowski's lemma on lattice points
   • Logarithmic spaces
   • Dirichlet's Unit Theorem for the units of the ring of integers of a number field
   • Theorems of Minkowski and of Hermite on discriminants of number fields
3. Ramification Theory:
   • Relative extensions
   • Relative discriminant and Dedekind's criterion for ramification in terms of discriminant
   • Higher ramification groups
   • Hilbert theory of ramification
4. Splitting of Primes:
   • Frobenius map
   • Artin symbol
   • Artin map
   • Splitting of primes in Abelian extensions in terms of Artin map
   • Rudimentary class field theory
   • Examples - quadratic and cyclotomic extensions
5. Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem
6. Dedekind zeta function
   • The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant

Textbooks

Possible textbooks for this course include, (but are not limited to):

1. • D. Marcus, Number Fields (Universitext)