Math 80220 Topics in Algebra (Spring 2018)
Algebraic Number Theory
When/where/who
- Andrei Jorza, ajorza, 275 Hurley Hall.
- Pasquerilla 109, MWF 9:25 - 10:15.
- Office hours: TBD.
Course Description
Modern number theory lies at the interplay between algebra, geometry, representation theory, and analysis (and even computer science) and the interactions among these fields led to numerous advances in the last half century. This course is an introduction to algebraic (and a little analytic) number theory: 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 Sage.
The first part of the course is devoted to understanding number fields and their rings of integers, including unique factorization into prime ideals and the group of units, using Galois theory and ramification theory. We will then study Minkowski's geometry of numbers for the finiteness of the class group, and Kummer's special case of Fermat's Last Theorem.
The second part of the course turns towards soft analytic methods: zeta functions, L-functions, the class number formula, Dirichlet's theorem on primes in arithmetic progression.
The last part of the course is set aside for special topics, depending on the students' preference. Possibilities include: harder analytic results (such as estimating the number of primes or exponential sums), primality testing and public key cryptography, number theory on function fields, elliptic curves mod p, etc.
Each topic will be accompanied by ample experimentation in Sage.
Prerequisites: Knowledge of rings, fields, and modules. Galois theory and complex analysis are useful but not prerequisites. Can take Galois theory in parallel.
Textbook: There is no official textbook for the course. You may find the following books useful:
- William Stein, Algebraic Number Theory, A Computational Approach (free).
- Marcus, Number Fields. (On reserve in the math library.)
Homework and Grades: There will be a weekly problem set and the grade will be based only on these problem sets.
The design of this webpage is based on the MIT course web page template.
