Course Description
Modern number theory lies at the interplay between algebra, geometry, representation theory and analysis and the interactions among these fields led to numerous advances in the last half century. This course is an introduction to algebraic number theory from a computational perspective: we will study the main objects (number fields and their rings of integers), their properties from a classical perspective, and their connections to algebra, geometry and analysis, while at the same time interact computationally with the main theorems of algebraic number theory in the open source computational algebra system Sage.
The main topics of the course will include: number fields and number rings; the class group and its structure using Minkowski's geometry of numbers; Kummer's special case of Fermat's Last Theorem; decomposition groups and ramification; the Dirichlet unit theorem; zeta functions and the class number formula (a special case of the million dollar Birch and Swinnerton-Dyer conjecture); L-functions and primes in arithmetic progression. Each topic will be accompanied by ample experimentation in Sage.
Prerequisites: Knowledge of rings and fields. Galois theory and complex analysis are useful but not prerequisites.
Textbook: William Stein, Algebraic Number Theory, A Computational Approach. In addition, Marcus Number Fields has been placed on reserve in the math library but it is not a textbook.
Homework and Grades: There will be a weekly problem set and the grade will be based only on these problem sets. You may choose to do, instead of one of the problem sets, a final project of your choice.
Announcements
January 15: I will be out of town during the week February 17 to February 21
