Instructor: Melanie Matchett Wood
Lectures
 TuTh 9:30AM  10:45AM in B231 Van Vleck.
The lecture schedule is posted below.
Office Hours
 Wed 9:30AM  10:45AM in 305 Van Vleck or by appointment.
Course Webpage: http://www.math.wisc.edu/~mmwood/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 oneyear 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
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).
The final version will be posted by the preceeding Thursday and noon and say FINAL at the top of the pdf. (Earlier versions will be posted if I decide some problems early.)
You may work together, but each student should do their own writeup of each problem. 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). 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.

Homework 1 (due Sep 9)

Homework 2 (due Sep 16)

Homework 3 (due Sep 23)

Homework 4 (due Sep 30)

Homework 5 (due Oct 7)

Homework 6 (due Oct 14)

Homework 7 (due Oct 21)

Homework 8 (due Oct 28)

Homework 9 (due Nov 4)

Homework 10 (due Nov 11)

Homework 11 (due Nov 18)

Homework 12 (due Dec 2note week skipped)

Homework 13 (due Dec 9)
 No more homework!
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.
As an introduction you could also look at the introduction and number fields sections of these
lectures on 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. (Also perhaps see his
documents on sage programming and using rings in sage.
Here is the
Sage reference manual (in particular the section on "Algebraic Number Fields" will be very useful).
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
 Sep 2: Introduction and integrality
 Sep 4: Properties of integralilty
 Sep 9: Discriminants
 Sep 11: Discriminants
 Sep 16: Dedekind domains
 Sep 18: Unique factorization of ideals in Dedekind domains
 Sep 23: Class groups of Dedkind domains
 Sep 25: Discrete valuations and extensions of Dedekind domains
 Sep 30: Factorization of prime ideals extensions of Dedekind domains
 Oct 2: Ramified primes
 Oct 9: Dedekind's criterion and norms of ideals
 Oct 14: Statement of Minkowkski's Theorem (and finiteness of class group) and lattices
 Oct 16: Geometry of numbers and sums of four squares
 Oct 21: Proof of Minkowksi's theorem
 Oct 23: Binary Quadratic Forms and Ideal Class groups of Quadratic fields
 Oct 28: Dirichlet Unit Theorem
 Oct 30: Dirichlet Unit Theorem
 Nov 4: Sunits, regulator, and cyclotomic fields
 Nov 6: Cyclotomic fields
 Nov 11: First case of Fermat's last theorem for regular primes
 Nov 13: Absolute Values
 Nov 18: Equivalent absolute values
 Nov 20: Completions with respect to absolute values
 Nov 25: Completions with respect to discrete absolute values, padics, Hensel's lemma
 Dec 2: Newton's method in the padics, extensions of discrete nonarchmedian absolute values on complete fields
 Dec 4: Extensions of discrete nonarchmedian absolute values on fields (complete or not), and correspondence with prime factorization in extensions
 Dec 9: Places of a number field or function field, extensions of absolute values and factorization of minimal polynomial in completion
 Dec 11: Product formula, decomposition and inertia groups
 Dec 16: Relationship of decomposition groups and inertia groups to completions