MA 844 A1 — Spring 2014
Algebraic Number Theory

Syllabus: The course syllabus is available here.

Homework: Problem sets will be assigned approximately every two weeks and posted below.

Student project: Each student will be asked to make a short presentation in class near the end of the semester going into greater depth of some topic covered in class. Alternatively, instead of a presentation, you may write a 5-10 page document (in LaTeX) on this topic. We will begin discussing precise topics halfway through the semester.

Course grading:  80% of your grade will be based on HW; 20% of your grade will be based on your student project.

References: There are dozens of possible references for algebraic number theory. Below are a list of a few that I draw upon for my lectures.

• Daniel Marcus, "Number Fields"

  A wonderful down-to-earth introduction to algebraic number theory with tons and tons of examples and exercises. This book, as suggested in the title, focuses on rings of integers in number fields. It's a great first book to work through -- don't let the type-setting scare you off.

• Tom Weston, "Algebraic Number Theory"

  These are course notes by Tom Weston on algebraic number theory. They are beautiful notes pitched at a level similar to Marcus' text full of tons of explanations of the intuition behind various constructions.

• James Milne, "Algebraic Number Theory"

  Well-known notes that are pitched at a higher level than the previous two treating more general rings of integers (not just the number field case). An excellent reference.

• Jurgen Neukirch, "Algebraic Number Theory".

  Classical reference now -- very general and thorough treatment.

• Keith Conrad's expository papers

  Many many beautifully written notes on a wide variety of topics relevant to this course.

Guide to lectures/references

1. Solving Diophantine equations, Fermat's last theorem and unique factorizations
   • Marcus Chapter 1
   • Weston Chapter 5
   • Conrad's "Mordell's equation" and "Fermat's last theorem for regular primes" notes.
2. Background for field theory
   • Marcus, Appendix 2
   • Weston, Appendix A
   • Conrad notes, too many to name
3. Rings of integers are rings and finitely generated (sometimes)
   • Marcus, Chapter 2
   • Weston, Chapter 2.2
   • Milne, Chapter 2 (not just over number fields)
   • Neukrich, Chapter 1.2 (ditto)
4. Unique factorization in Dedekind domains
   • Marcus, Chapter 3
   • Weston, Chapter 2.3
   • Milne, Chapter 3
   • Neukrich, Chapter 1.3
5. Geometry of numbers (finiteness of class groups/Dirichlet Unit theorem)
   • Marcus, Chapter 5
   • Neukrich, Chapter 1.4-1.7
6. Extensions of Dedekind domains and explicit factoring of prime ideals
   • Marcus, Chapter 3
   • Neukrich, Chapter 1.8
7. Factoring and Galois theory and QR
   • Marcus, Chapter 4
   • Neukrich, Chapter 1.9
8. Cyclotomic fields
   • Marcus, within Chapters 2 and 3
   • Neukrich, Chapter 1.10
   • Weston, Chapters 1.3, 2.4