**
MATH 681 Spring 2018
**

# Basic Information

**Instructor**: Jeff Thunder**Office**: WH 362**Phone**: 753-6761**e-mail**: jthunder@math.niu.edu**Office Hours**: Monday, Wednesday, Friday, 11:00-11:50 a.m. or by appointment.

# Text and Syllabus

The textbook for the course is *Number Fields*
by Daniel A. Marcus. We will attempt to cover as much of the material
from the first seven chapters as is reasonable.

# Grading Scale

Grades for section 1 will be based on homework and the final exam. The weights for these are 70% and 30%, respectively.

# Homework

Homework will be assigned somewhat irregularly. I'll announce assignments (and due dates) in class and also post them to this webpage. You are free to work with other students on the homework; in fact, this is encouraged. Sloppy and/or illegible work will be returned back with no credit! Your homework is something of which you should be proud (notice how I didn't end with a preposition there). Expect to spend lots of time on it.

# Homework Assignments

- First homework assignment (Due 1/23)
- Second homework assignment (Due 1/30)
- Third homework assignment (Due 2/6)
- Fourth homework assignment (Due 2/13)
- Fifth homework assignment (Due 2/20)
- Sixth homework assignment (Due 2/27)
- Seventh homework assignment (Due 3/29)
- Eighth homework assignment (Due 4/19)
- Ninth homework assignment (Due 5/1)

# Final Exam

The final exam will be held Tuesday, May 8 from 2:00-3:50 in DuSable 252.

# Handouts

- I've TeXed up a compilation of some useful results from algebra (mainly from MATH 620).
- Prologue (stuff we talked about the first two days of class)
- Basic stuff about number fields and algebraic integers discussed Jan. 23 in class.
- A general result on lattices with an application to number fields.
- The Fundamental Theorem of fractional ideals.
- This handout contains goodies such as the definition of order at a prime ideal and the Chinese Remainder Theorem. Good stuff.
- I've written up a proof of Dedekind's theorem for your reading enjoyment. Homomorphisms, kernels, commutative diagrams - the budding algebraist in you squeals with delight.
- A brief introduction to some geometry of numbers.
- Some applications of Minkowski's Theorem.
- And another.
- Dirichlet's unit theorem.
- The Dedekind-Weber Theorem.
- Basic information on the zeta function of a number field.
- Some classical results on the different of a number field.
- Notes on absolute values and the places of Q.
- Notes on absolute values places of number fields and function fields.
- Notes on heights on number fields and function fields.
- The Riemann-Roch Theorem and sundry applications thereof.
- Notes on Integral Closure and Complementary Modules. Always compliment your complementary modules!
- Dedekind's different theorem.

## Homer does math!

Yes indeed, there's plenty of math humor to be found in the Simpsons. Just look and see!

Last update: May 3, 2018