Math 254A - Number Theory

Instructor: Brian Osserman

Lectures: MWF 11:00-12:00pm, Room 65 Evans

Course Control Number: 55182

Office: 767 Evans, e-mail:

Office Hours: M 3:00-4:00pm, W 1:00-2:00pm, F 2:00-3:00pm

Prerequisites: Math 250A required, 115 and 250B recommended

Required Text: Lang, Algebraic Number Theory

Recommended Reading: Cox, Primes of the form x^2+ny^2

Syllabus: Roughly, we will cover algebraic number theory from scratch, using Lang's book with examples and motivation from other sources. Advanced topics such as class field theory and Tchebotarev density will be discussed with an emphasis on the statements and applications, so proofs of these will be omitted, and they will be treated in the classical language of ideals rather than ideles and group cohomology.

Grading: Homework: 2/3rds of grade, final paper: 1/3rd of grade.

Homework: Homework will be assigned weekly, on Mondays, except that they will be assigned the previous Friday when Mondays are holidays.

Final paper guidelines

The purpose of the final paper is twofold: to learn in detail about a topic not covered in class, and to practice mathematical writing. As such, it will be graded as much on clarity of writing as on any other factors.

Guidelines:

The paper should be roughly 8-10 pages in default tex formatting. Using latex with the amsart documentclass is recommended, but if you are already familiar with another flavor of tex, you may use that instead.
There is no requirement of originality, but of course the paper should be entirely in your own words.
The paper should be fully self-contained, and written like a research article. In particular, it should have a brief introduction explaining the main material to be discussed, and ideally some explanation of why it is important.
Material from lecture may be used without citation. Material closely related to but not covered in lecture should be used with a specific statement of the result needed, and a citation. Many of you will also have to use results from other fields. These should always include a specific statement and citation, and you should consult with me on which results you are quoting.

Lecture notes

I will attempt to type up and post lecture notes. If I have not prepared notes for a lecture, I will give advance warning. Actual problem sets are posted below, so feel free to ignore exercises mentioned in the lecture notes.

8/29 1: Motivational lecture (PDF)
8/31 2: Rings of integers (PDF)
9/02 3: Rings of integers and Dedekind domains (PDF)
9/07 4: Dedekind domains I (PDF)
9/09 5: Dedekind domains II (PDF)
9/14 6: The ideal class group, and unit theorem (PDF)
9/16 7: The ideal class group (PDF)
9/19 8: Examples of Minkowski bounds (PDF)
9/21 9: Minkowski bounds, and the unit theorem (PDF)
9/23 10: The unit theorem I (PDF)
9/26 11: The unit theorem II (PDF)
9/28 12: Factorization of prime ideals in extensions (PDF)
9/30 13: The discriminant and ramification (PDF)
10/03 14: Discriminants and applications (PDF)
10/05 15: Explicit factorization (PDF)
10/07 16: More on rings of integers (PDF)
10/10 17: Cyclotomic fields and Fermat's Last Theorem (PDF)
10/12 18: Fermat's Last Theorem, and generalized zeta functions (PDF)
10/14 19: The analytic class number formula I (PDF)
10/17 20: The analytic class number formula II (PDF)
10/19 21: The analytic class number formula, and Dirichlet characters (PDF)
10/21 22: Dirichlet L-series (PDF)
10/24 23: Prime factorization and Galois groups (PDF)
10/26 24: Abelian extensions (PDF)
10/28 25: Sub-cyclotomic fields (PDF)
10/31 26: Zeta functions and L-series (PDF)
11/02 27: Evaluating the L-series (PDF)
11/04 28: Introducing local fields (PDF)
11/07 29: Local fields II (PDF)
11/09 30: Local fields III (PDF)
11/14 31: Local fields IV (PDF)
11/16 32: Class field theory: an overview (PDF)
11/18 33: Class field theory II (PDF)
11/21 34: Class field theory III (PDF)
11/23 35: Hilbert class fields (PDF)
11/28 36: Ring class fields and p=x²+ny²(PDF)
11/30 37: Density theorems (PDF)
12/02 38: Density theorems via class field theory, and Fermat's Last Theorem revisited (PDF)
12/05 39: Interlude on Kummer theory (PDF)
12/05 40: Fermat revisited (PDF)
12/05 41: Bernoulli numbers and FLT (PDF)

Problem sets

9/02: Problem Set 1, due 9/12 (PDF)
9/12: Problem Set 2, due 9/19 (PDF)
9/19: Problem Set 3, due 9/26 (PDF)
9/26: No problem set this week
10/03: Problem Set 4, due 10/10 (PDF)
10/10: Problem Set 5, due 10/24 (PDF)
10/24: Problem Set 6, due 10/31 (PDF)
10/31: Problem Set 7, due 11/07 (PDF)