Math 373/573 Algebraic Number Theory Semester: Spring 2019

Description of course: The main object of study in Algebraic Number Theory are number fields (finite extensions of the field of rational numbers) and their rings of algebraic integers (those elements that are roots of monic polynomials with integer coefficients). Rings of algebraic integers have a theory of prime ideals that encode many interesting properties about the usual rational prime numbers, for example, the decomposition of primes in the Gaussian integers detects whether they can be written as a sum of two squares (like 5) or not (like 7). One of the most fundamental algebaic invariants of a number field is its ideal class group, which measures, among other things, the failure of unique factorization in the ring of integers. One of the fundamental theorems in Algebraic Number Theory is the finiteness of the ideal class group. Another is about the structure of units in the ring of integers. Both of these fundamental invariants are also mysteriously hidden in the Dedekind zeta function of a number field, analogous to the classical Riemann zeta function.

The topics covered will include: the ring of algebraic numbers; decomposition of ideals into products of prime ideals; the discriminant, different, and ramification theory; finiteness of the ideal class group; Dedekind unit theorem; the Dedekind zeta function; the p-adic numbers, ring of adeles, and group of ideles.

Expected background: Previous exposure to abstract algebra, field and Galois theory, and some complex analysis is required. For example, the contents of Math 350 Introduction to Abstract Algebra, Math 370 Fields and Galois Theory, and Math 310 Introduction to Complex Analysis, would suffice.

Work with anyone on solving your homework problems,
but write up your final draft by yourself.