Math 530, Algebraic Number Theory
Spring 2009
Course Description
This graduate course provides an introduction to algebraic number
theory, that is, the study of finite extensions of the rational
numbers and their rings of integers (typical example: the field
Q(i) and its subring Z[i] of Gaussian
integers). In particular, I will further develop the theory of fields
and rings of integers, including topics from ideal theory, units in
algebraic number fields, ramification, valuation theory,
local-to-global principles, function fields, and local class field
theory. Applications will include explaining quadratic reciprocity
and understanding quadratic forms over Q.
Here is the standard syllabus
for this course.
Prerequisites: Math
500 or a similar graduate-level abstract algebra class.
Grading
Your course grade will be based on
- Weekly homework assignments: (20%) These will
typically be due on Wednesday. Late homework will not be
accepted; however, your lowest two homework grades will be
dropped so you are effectively allowed two infinitely late
assignments. Collaboration on homework is permitted, nay
encouraged. However, you must write up your solutions
individually and understand them completely.
- Midterm Exam: (30%) The midterm exam will be Monday,
March 9. This will be a 2-hour exam, held in our usual
classroom, AH 341. You can choose to take the exam either from
8am-10am or from 9am - 11 am.
- Final Exam: (50%) This will be Thursday, May 14 from 8-11am.
Textbook
The required text for this course is:
- Daniel A. Marcus, Number Fields, Springer Universitext Series, ISBN: 0387902791.
though some additional topics will be drawn from other sources,
such as Neukirch's Algebraic Number Theory and Serre's A course in arithmetic.
Homework Assignments
- HW #1. Due Friday, January 30.
- HW #2. Due Wednesday, February 11.
- HW #3. Due Wednesday, February 18.
- HW #4. Due Wednesday, Febuary 25. Marcus, Chapter 3: #14, 16, 17, 24, 33.
On #33(c), do any 5 subparts.
- HW #5. Due Wednesday, March 4.
- HW #6. Due Friday, March 20.
- HW #7. Due Wed, April 8.
- HW #8. Due Wed, April 15.
- HW #9. Due Wed, April 22.
- HW #10. Due Wed, April 29.
- HW #11. Due Wed, May 6.
Lecture notes
Here scans of my lecture notes, in PDF format.
- Jan 21: Introduction: the Gaussian integers.
- Jan 23: Algebraic integers; norm and trace.
- Jan 26: The discriminant.
- Jan 28: Additive structure of the ring of integers.
- Feb 2: Restoring unique factorization: overview.
- Feb 4: Unique factorization for Dedekind domains.
- Feb 6: The ideal class group.
- Feb 9: The norm of an ideal.
- Feb 11: Integral bases for cyclotomic fields.
- Feb 13: Primes in extensions.
- Feb 16: Proof of the Fundamental Identity.
- Feb 18: Ramified primes divide the discriminant.
- Feb 20: Hilbert's ramification theory I.
- Feb 23: Hilbert's ramification theory II.
- Feb 25: Computing prime decompositions.
- Feb 27: Quadratic Reciprocity and cyclotomic fields.
- Mar 2: Primes in cyclotomic fields.
- Mar 4: Proof of quadratic reciprocity; generalizations.
- Mar 11: Frobenius automorphism; ramified primes; geometry of numbers.
- Mar 13: Minkowski theory I.
- Mar 16: Minkowski Theory II.
- Mar 18: Class groups are finite; Minkowski lattice point theorem.
- Mar 20: The units of integer rings I.
- Mar 30: The units of integer rings II.
- Apr 1: The units of integer rings III.
- Apr 3: p-adic numbers I.
- Apr 6: p-adic numbers II.
- Apr 8: Solving equations with p-adic numbers.
- Apr 10: Hensel's Lemma; Places and primes.
- Apr 13: Quadratic forms I.
- Apr 15: Quadratic forms II.
- Apr 17: The Hilbert symbol.
- Apr 20: Quadratic forms over Q_{p}.
- Apr 22: Hasse-Minkowski Theorem.
- Apr 24: Hilbert's symbol revisited.
- Apr 27: Local to global for Hilbert's symbol I.
- Apr 29: Local to global for Hilbert's symbol II; The Approximation Theorem.
- May 1: Hasse Minkowski Revisited.
- May 4: Hasse Minkowski fin; Adeles and Ideles.
- May 6: Geometry, topology, and
number theory.
Recommended article: Matthew Emerton, Topology, representation theory, and
arithmetic: 3-manifolds and the Langlands Program.