Math
81/111 Abstract Algebra (Rings and Fields)
Winter 2016
Course Info:
Lectures: Monday,
Wednesday, Friday, block 10 (10:00am – 11:05am)
x-hour: Thursday
12:00pm-12:50pm
Room: Kemeny
004
Instructor: Sam
Miner
Email: samuel.a.miner@dartmouth.edu
Office: Kemeny
318
Office hours: Tuesday 1-2pm, Thursday 10-11am, or by appointment
Course Webpage: http://www.math.dartmouth.edu/~m81w16/
Prerequisites: Math 71, or Math 31 and instructor permission
Required Texts: J.S. Milne, Fields and Galois Theory,
version 4.51
Recommended Texts: Dummit
and Foote, Abstract
Algebra
Lang, Algebra
Grading:
Grading will be based on
weekly homework (30%), a midterm exam (30%), and a final exam (40%).
Course Description:
This course provides a
foundation in core areas in the theory of rings and fields. Specifically, it
provides
an introduction to commutative ring theory with a
particular emphasis on polynomial rings and their
applications to unique factorization and to finite and algebraic
extensions of fields. The study of fields
continues with an introduction to Galois Theory, including the fundamental
theorem of Galois Theory
and numerous applications.
Homework:
Homework assignments will be
assigned weekly, and will be due on Mondays (except for HW#1).
Collaboration is encouraged,
though you must each write up your own assignment. Please acknowledge
any cooperation at the end of each assignment.
As always, the Honor
Principle applies to your work in this course.
Course Schedule:
|
1 |
Jan 11 |
(M) |
Introduction |
|
|
2 |
Jan 13 |
(W) |
Ideals, domains, Chinese
Remainder Theorem |
|
|
3 |
Jan 14 |
(R) |
Xhour: Existence of maximal ideals, EDs, PIDs |
|
|
4 |
Jan 15 |
(F) |
Divisibility,
irreducibility and primes, UFDs |
|
|
- |
Jan 18 |
(M) |
No class (MLK Day) |
|
|
5 |
Jan 20 |
(W) |
UFDs are PIDs,
factorization in polynomial rings |
HW #2 (due Jan 25) |
|
6 |
Jan 21 |
(R) |
Xhour: If R is a UFD, R[X] is a
UFD, EisensteinŐs criterion |
|
|
7 |
Jan 22 |
(F) |
Extension fields |
|