Instructor: Melanie Matchett Wood
Lectures
Office Hours
Course Webpage: http://www.math.wisc.edu/~mmwood/748Fall2016/748.html
Course Description: An introductory graduate level course on algebraic number theory. Topics: a rigorous introduction to the arithmetic of number fields, unique factorization, algebraic integers, Dedekind domains and factorization of ideals, geometry of numbers, Dirichlet's Unit Theorem, ideal class groups, first case of Fermat's Last Theorem, local fields.
Prerequistes:
a one-year course on Abstract Algebra at the graduate level, including various standard facts about groups, rings, fields, vector spaces, modules, and Galois Theory.
Text: Milne's Algebraic Number Theory Notes
Quick links
Grades
Grades will be based on homework.
Homework assignments:
Homework assignments will be due each Tuesday at the start of class (paper copies must be handed in). Homework will not be accepted late (this is for your benefit so you keep up with the lectures).
You may work together to solve the problems, but each student should do their own writeup of each problem and you should write for each problem which other students you worked with on it. The problems from Milne have hints and/or solutions in the back of the text (though obviously for your own sake, you should do the problems without reference to these as much as possible). Some of the homework will require sage computations (see below) . For the sage problems, please print some readable version of your sage terminal window, worksheet, or code and output, and indicate with written notes what the answer is.
Sage:
Sage is open source mathematics software that is, in particular, commonly used by number theorists.
You can find a tour, tutorial, and further introductory material on sage on the sage website.
Here is an introductory video on using sage for algebraic number theory.
Here is a written
introduction to using sage for number theory .
There are lots of ways to use sage: in a terminal, in a browser worksheet, in sagemathcloud. You can pick whatever works best for you.
Lalit Jain has created a nice Sage/Linux quickstart reference which is available here.
Here is the Sage reference manual (in particular the section on "Algebraic Numbers" will be very useful).
Other texts:
In this course, we will follow Milne's notes closely. There are many other great resources for learning algebraic number theory, and you might also use a complementary text if you find you need more explanation, or examples, or just another point of view. Some books I recommend include Marcus's ``Number Fields'' and Janusz's ``Algebraic Number Fields.'' Lang's ``Algebraic Number Theory'' and Neukirch's ``Algebraic Number Theory'' are also standard references.
Some other online notes include