Math 358: Introduction to Algebraic Number Theory, Spring 2006

Instructor:

Chad Schoen

Time:

MWF 1:30 - 2:20

Location:

Room 205 Physics Building. (Note this is the NEW room number.
Until December 31 2005 this room was numbered 228E and that number may still be on the door.)

Homework discussion period:

Tuesdays 9:00-9:50 am in room 205 physics.

Prerequisites:

Math 251. Students will be expected to enter the course with a working knowledge of the following basic notions in algebra, all of which are treated in Math 251: Very basic group theory, actions of groups on a set, commutative rings, ideals, modules over commutative rings, principal ideal domains, unique factorization domains, the classification of finitely generated modules over principal ideal domains, field extensions, the structure of finite fields, Galois theory.

About the course:

The material covered in this course is of fundamental importance in the fields of algebra, algebraic geometry, and number theory. A graduate level course in number theory is typically offered once every two years at Duke. However the subject matter varies greatly from course to course. The course in spring 2006 concentrates on algebraic number theory and is related to basic commutative algebra and to foundational material in algebraic geometry. In other years this course has concentrated on zeta functions or on modular forms, topics which are closely related to complex analysis. It is likely that a course with a similar algebraic emphasis will not be offered within the next 3 years.

Intended audience:

Graduate students at any level who have completed math 251 and who are considering working in algebra, algebraic geometry or another area of mathematics which interacts with number theory. Any student, undergraduate or graduate, who has met the prerequisite and is interested in expanding his/her general mathematical knowledge by learning the core material in algebraic number theory.

Level of difficulty:

This course will probably be more demanding than Math 251.

Relationship with other graduate level courses

Math 251: Math 358 may be regarded as a sequel to Math 251.

Math 252: There will be some overlap with the commutative algebra course, but not a great deal. Thus it is possible to take math 358 before taking math 252, after taking math 252, while taking math 252 concurrently, or without taking math 252 at all, although the last option is not recommended for students who intend to write a PhD thesis in algebraic geometry at Duke. The commutative algebra required for number theory is easier and more concrete than the commutative algebra covered in Math 252 and required for algebraic geometry. This is because the rings which appear in number theory have dimension at most one, whereas those studied in commutative algebra and algebraic geometry have arbitrary dimension. The difference is like the difference in geometry between curves and higher dimensional objects.

Math 272: Riemann surfaces are one dimensional complex manifolds. Math 272 is primarily concerned with compact Riemann surfaces. These all turn out to come from algebraic curves over the complex numbers. A main theme in the Riemann surface course is to prove this fact. Math 358 is very closely related to the theory of algebraic curves over arbitrary fields and especially to algebraic curves over finite fields. This subject finds applications in the theory of error correcting codes. There will definitely be connections between Math 358 and Math 272. Precisely how strong these connections are depends upon how much time is available to discuss algebraic curves in Math 358. Despite the connections, there will be very little overlap as Math 272 takes its techniques from complex analysis with a little functional analysis and algebraic topology thrown in, while Math 358 will use completely algebraic methods.

Math 273: Math 358 should complement the introductory algebraic geometry course nicely. There will be little overlap. The core material in algebraic number theory turns out to be fundamental for work in algebraic geometry beyond Math 273.

Courses in elliptic curves and related topics: From timte to time the math department has offered such an advanced course. Math 358 would provide the number theory background needed for such a course.

More advanced courses in algebraic geometry: Modern research in algebraic geometry is based on the notion of a scheme. This notion is not used in the standard algebraic geometry course, Math 273, because on can get much further much faster with the more down to earth notion of a clasical algebraic variety. The true power of schemes in algebraic geometry is only appreciated after the first course in the subject. By contrast schemes give an immediate payback in algebraic number theory by offering a geometric point of view on the subject which is not otherwise accessible. Furthermore the schemes involved are quite elementary. Math 358 will include a gentle introduction to schemes.

Topics:

The course aims to cover the core topics in an introductory one semester course in algebraic number theory and to supplement these as time permits with closely related topics in the field of algebraic curves.

Quadratic equations in two variables, Number fields and and a tiny bit of function fields, orders in number fields and a tiny bit of algebraic curves over finite fields, the spectrum of a ring, integral closure and resolving singularities, Dedekind domains, the class group, lots of examples of all the above, the geometry of numbers, finiteness of the class group and Dirichlet's unit theorem, decomposition of primes in extension fields, p-adic numbers, the Hasse principal for ternary quadratic forms, valuation theory, the different, ramification theory, finiteness of extensions of number fields of bounded degree unramified outside a fixed finite set.

Text:

Neukirch, J.; Algebraic Number Theory

Additional references:

Some additional references to be placed on reserve in the library:
Baker, A Concise Introduction to the Theory of Numbers

Grading:

Grading will be based primarily on homework. Homework may be
assigned to write up and turn in. It may be assigned simply to discuss in a weekly
homework discussion session or it may be assigned to present to the class in a
weekly homework discussion session. Poor attendance at class or at the weekly
discussion sessions could result in a lower grade. No tests are contemplated.

Table of positive definite binary quadratic forms of small discriminant.

Homework:

1. Homework for discussion at 9am on Tuesday, January 17 in room 205 Physics Building.
Topics: Representing integers by binary quadratic forms. Reduction theory.
Reading: The first three sections of chapter 5 of Baker, A Concise
Introduction to the Theory of Numbers, (pages 35-38). This book is on
reserve for this course in Vesic library.
Problems: Handout: 1. Binary Quadratic Forms.
Please try these problems. Formulate questions on those where you run into difficulty.
Be prepared to discuss these problems on Tuesday, January 17, 2006 9:00-9:50 in room 205.

2. Homework for discussion at 9am on Tuesday, January 24 in room 205 Physics Building.
Topics: Integers in quadratic number fields. Connection with binary quadratic forms.
Reading: Artin 11.6, 11.7
Problems:
1. Look at the table which we have constructed. Try to guess for which
negative fundamental discriminants, D, the class number, h(D), is even.
Check your guess by filling in two missing lines in the table and e-mailing
me these missing lines for inclusion.
2. Compare Artin Theorem 7.7 with the table we have constructed. What relationship
do you notice?
3. Handout: 2. Orders in quadratic fields and binary quadratic forms
4. Artin section 11.6: 7,8.
5. Artin section 11.7: 3, 8, 9.

3. Homework for discussion on Tuesday January 31 at 9am.
Topics: Unique factorization of ideals in the integers of
quadratic number fields. Class groups. Computation of class groups
in the case of imaginary quadratic fields.
Reading: Artin 11.8-9 and 11.10 through Corollary 10.7.
Problems to look at:
Handout: 3. Binary quadratic forms and ideal class groups.
Artin section 11.8: 3.
Artin section 11.9: 1a.

4. Homework for discussion on Tuesday February 7.
Topics: Integrality, integrally closed integral domains, algebraic integers,
the trace form, discriminants.
Reading: Neukirch I.2 (ie. Chapter one, section 2)
Problems for discussion:
Artin section 11.9: 1b.
Neukirch I.2: 1, 3, 5, 6.
Handout: 4. Integral extensions of rings and assorted problems.

5. Homework for discussion on Tuesday February 14.
Topics: Dedekind domains. Geometry of numbers.
Reading: Neukirch I.3,4,5
Neukirch I.2: 7.
Neukirch I.3: 1,2,3,4,5,6.
Handout: 5. Computations of rings of integers and assorted problems.
Handout: 7. Geometry of numbers.

6. Homework for discussion on Tuesday February 21.
Topics: Finiteness of the ideal class group. Computing class groups.
Number fields with fixed discriminant.
Reading: Neukirch I.6, III.2.14-17.
Optional reading: Artin section 11.10
Handout: 6. Class numbers of number fields

7. Homework for discussion on Tuesday February 28.
Topics: Dirichlet's unit theorem.
Reading: Artin section 11.11.
Reading: Neukirch I.7.
Exercises: Neukirch I.7: 1-6.

8. Homework for discussion on Tuesday March 7.
Topics: Reduction theory of indefinite binary quadratic forms and Pell's equations.
Optional reading in Chapter 4 of Flath on reserve in library.
Handout: 8. Reduction of indefinite quadratic forms

9. Homework for discussion on Tuesday March 21.
Topics: Localization and extension of Dedekind domains.
Decomposition of prime ideals in extensions of Dedekind domains.
The p-adic numbers (reading only).
Reading: Neukirch I.11 through the proof of theorem 11.4.
Reading: Neukirch I.8 through the proof of theorem 8.4.
Reading: Neukirch I.9
Exercises: The main exercises are on Handout 10, but other possibilities are listed as well.
Exercises: Neukirch I.11: 6.
Exercises: Neukirch I.9: 1.
Handout 9. Localization.
Handout 10. Decomposition of primes.

10. Homework for discussion on Tuesday March 28.
(Topics: Decomposition of primes in cyclotomic, quadratic and cubic fields, Quadratic Dirichlet characters.) Reading: Neukirch I.10
Exercises: Neukirch I.9: 2.
Exercises: Neukirch I.10: 2,3,4,5.
Handout 11.

11. Homework for discussion on Tuesday April 4.
(Topics: Representation of integers by bionary quadratic forms, splitting of primes
in quadratic fields, genus theory.)
Reading: Roughly first two pages of I.13 in Neukirch (ie. read through first example).
Handout 12.

12. Homework for discussion on Tuesday April 11.
(Topics: Spectra of rings. Singularities in dimension one and normalization.
Constructions of rings of integers.)
Reading: The beginning of Neukirch I.12 (through the proof of Theorem 12.2).
Reading: Neukirch I.13.
Exercises: Neukirch I.12: 1.
Handout 13. Problems in order of decreasing priority:
1, 2, 3, 13,10, 4, 5,6,8,9,11,12.

13. Homework for discussion on Tuesday April 18.
(Topics: p-adic numbers, I-adic completion of a ring, Hensel's Lemma.)
Reading: Neukirch II.1 and II.2 and II.4.6 (Hensel's Lemma).
Handout on p-adic numbers:1-4, 16-28.
Handout 13: 11, 12(part i, ii, or iii, your choice).
Handout 14. Problems in order of decreasing priority:1,2,3,7.

14. Homework for possible discussion next week (but not at 9am Tuesday).
(Topics: Hasse principal for ternary quadratic forms.)
Reading: Neukirch II.3 and II.4 (skip theorem II.4.2).
Reading (optional): Neukirch II.7-9. We cover some of this material
although our treatment is less extensive.
Reading: Handout on valuation theory.
Exercises: Handout 15.

WARNING: WHAT FOLLOWS IS UNDER CONSTRUCTION AND WILL LIKELY CHANGE SHORTLY.

15. Homework for idle moments in the summer.
Topic: The different and the precise relationship between the discriminant and ramification.
Reading: Neukirch III.2
Exercises: Neukirch III.6: 2. Neukirch III.7: 1,3

Theorem List:
Finiteness of the class group of a number field.
Dirichlet's unit theorem.
Minkowski's theorem (non-existence of non-ramified extensions of Q).
Finiteness of the number of isomorphism classes of number field with fixed discriminant.
Artin reciprocity for cyclotomic extensions.
Quadratic reciprocity.
Reduction theory of definite binary quadratic forms.
Reduction theory of indefinite binary quadratic forms.
Algorithm for solving Pell's equation.
Proof that every natural number is a sum of four squares.
Legendre's theorem (Hasse principle for ternary quadratic forms).
Finiteness of the number of number fields of bounded degree unramified outside a finite set of places.