Math 358: Introduction to Algebraic Number Theory

Instructor:

Time:

Prerequisites:

About the course:

Intended audience:

Relationship with other graduate level courses

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 function fields, orders in number fields and algebraic curves over finite fields, affine schemes, 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, valuation theory, ramification theory, extensions of global fields with fixed ramification, bounds on discriminants. Further topics as time permits.

Text:

Additional references:

Some additional references to be placed on reserve in the library:
Lang, Algebraic Number Theory
Samuel, Introduction to the Algebraic Theory of Numbers