Math 5230 - Fall 2016
Algebraic Number Theory

Instructor Keith Conrad
Email kconrad at math dot uconn dot edu.
Office hours 11:00-12:00 M, 2:30-3:30 W, or by appointment in MONT  234.
Course info
Lecture MWF 1:25-2:15 in MONT 321.
 
Texts Algebraic Theory of Numbers by Pierre Samuel.

Other lecture notes on algebraic number theory by James Milne (to get his notes, look in the left margin under Course Notes for the title Algebraic Number Theory), René Schoof (2003), Peter Stevenhagen (2012), and Tom Weston (1999) may be helpful, but are not required.

Course Handouts

PARI: an introduction and some commands specific to algebraic number theory. A reference card.

The Gaussian Integers.

Examples of Mordell's equation.

Unique ideal factorization in quadratic fields.

The trace and norm for field extensions: part I and part II.

Modules over a PID. Read especially Section 4 about the generalization of cardinality and index beyond the setting of modules over Z.

A non-free relative integral extension.

Unique ideal factorization in rings of integers.

Factorization of ideals by Dedekind's method.

Ideal class group calculations using Kronecker's bound.

Applications of ideal class groups: relative extensions of integers, orbits of SL2, and matrix conjugation over Z.

Discriminants and ramified primes.

Totally ramified extensions and Eisenstein polynomials.

Irreducibility of truncated exponential series. (optional)

Class Group Calculations.

Pell's equation (units in quadratic fields): part I and part II.

Dirichlet's Unit Theorem.

Play Lucy and Lily, an algebraic number theory game. To read about it, look here.

Existence of Frobenius elements.

Recent Announcements

8/29: First lecture.

Brief course description: This course is an introduction to algebraic number theory: rings of integers, applications to diophantine equations, ramification, Dirichlet unit theorem, ideal class group, and Frobenius elements in Galois groups. Further topics (e.g., local fields or zeta-functions) will be discussed if time permits.

Prerequisites: Math 5211.

Course grade:  This will be based on homeworks and a take-home final.

Homework: Homework assignments will be posted on the bottom of this web page.



Due Week of Homework Assignment
1. Aug. 29
2. Sept. 5

3. Sept. 12
Set 1.
4. Sept. 19

5. Sept. 26
Set 2.
6. Oct. 3

7. Oct. 10
Set 3.
8. Oct. 17
9. Oct. 24
Set 4.
10. Oct. 31

11. Nov. 7
12. Nov. 14
Set 5.
13. Nov. 21
None (it's Thanksgiving).
14. Nov. 28
15. Dec. 5

16. Dec. 12
Problem Set in Lieu of Final. Do NOT discuss the problems with anyone except the instructor.