Math 525: Algebraic Number Theory
This is the information page for Math 525, Fall 2002.
Text: A Classical Introduction to Modern Number Theory, 2nd edition,
by K. Ireland and M. Rosen. Supplemental material on algebraic number
fields will be drawn from "Algebraic Number Theory" by Stewart
and Tall.
I do not expect to cover the complete text in a single semester.
Instead I plan to discuss Chapters 5-12,
with supplemental material on algebraic number fields as time
permits. Many of the standard results in algebraic number
theory can also be regarded as results in algebraic K-theory,
so algebraic topologists may be interested in this course.
Homework
| Assignment |
Chapter |
Problems |
Date Due |
| 1 |
5 |
1, 6, 10, 14, 16, 22, 23, 24, 25 |
September 10 |
| 2 |
6 |
1,2,3,4,5,6,7,23 |
September 19 |
| 3 |
6 |
11, 15, 18 |
|
|
7 |
6, 7, 8, 18, 21, 22, 23 |
September 26 |
| 4 |
8 |
3, 4, 5, 9, 11, 12 |
Thursday, October 10 |
| 5 |
8 |
13, 16, 17, 18, 27 |
Thursday, October 17 |
| 6 |
9 |
2, 3, 4, 5, 6 [(a) only], 7 [-7-3w only], 8 [143 only], 13, 14 |
Tuesday, October 29 |
| 7 |
9 |
27, 28, 29, 30, 33, 45 |
Thursday, November 7 |
| 8 |
handout |
(8 problems) |
Tuesday, November 26 |
| 9 |
handout |
(7 problems) |
Thursday, December 12 |
Tests
| Test |
Material |
Location |
Date and Time |
| 1 |
Chapters 5-7 |
LN 2205 |
October 2, 7-9 PM |
| 2 |
Chapters 8-9 |
LN2205 |
November 13, 7-9PM |
| 3 |
|
|
|
| Final Exam |
Entire Course |
SW325 |
December 17, 2-4 PM |
What to know for the final exam:
- Statement, proof, and applications of quadratic reciprocity. The
proof to know is the second one, which is really the same
as the third one. You need not learn the details of the
first proof.
- Basic properties of finite fields, Gauss and Jacobi sums. For
example, you may ignore Jacobi sums involving more than
2 characters.
- Properties of the rings Z[w] and Z[i]: units, primes, quotient
fields, and norms.
- Statement and application of cubic and biquadratic reciprocity.
Of course this includes the definition of the relevant
reside symbols. You need not know the proofs of these
results.
- General facts about the ring of integers in an number field. For
example, you should be able to prove that the ring
of integers, considered as an additive abelian
group, is free of rank equal to the degree of the
extension.
- Quadratic and cyclotomic fields: know what the rings of integers
are, and why. You need not memorize the
discussion of the ring of integers in cyclotomic
fields, but you should be able to, for example,
compute the discriminant.
- Unique factorization of ideals: Know the result and its
applications. You need not learn the 9-step
proof.
- Applications to solving equations when unique factorization is
known. For example, assuming unique
factorization in the relevant ring, you should
be able to show that the only positive integer
solutions of y^2 + 4 = x^3 are x = 5, y = 11
and x = 2, y = 2. (This is a fairly
time-consuming problem, and were it on the
test, it would be packaged in parts, with hints.)
- You need not know the statement, proof, or applications of
the Eisenstein Reciprocity Law.
Note that the emphasis in the exam will be on the material since the
second test.