Instructor: Amod Agashe

Textbook: Lang's "Algebraic number theory" (Chapters I, III, IV, V, and II, in that order) and/or Janusz's "Algebraic number fields" (Chapters I and II).

Description: This will be a standard beginning course in algebraic number theory. Algebraic number theory was one of the motivations for the initial developments in commutative algebra and it also has some parallels with algebraic geometry. Hence the course should be useful for graduate students in algebra in general.

The primary object of study of algebraic number theory is the ring of algebraic integers. We will show that these rings are Dedekind domains, and thus one has unique factorization into prime ideals. The failure of factorization into prime elements is measured by the ideal class group, and we will show that this group is finite. We will also study the structure of the unit group. We will develop techniques to compute the ring of integers and give applications to problems concering the usual integers, e.g., quadratic receprocity, as well as to cryptography, especially to primality and factoring.

Topics to be covered include: rings of integers, Dedekind domains, prime factorization, discriminants, ramification, finiteness of the class number, Dirichlet unit theorem, cyclotomic extensions, quadratic reciprocity, completions and valuations, Hensel's lemma, and other topics if time permits.

Prerequisite: a basic graduate course in algebra (e.g., Math 8210 Basic Algebra should be enough).