MATH 7370: Algebraic Number Theory (Fall 2010)

Prerequisites: MATH 4340 or equivalent.

This course is a basic introduction to algebraic number theory. A core topic deals with the ideal theory of Dedekind domains as applied to the rings of integers of number fields (finite extensions of Q). A major purpose of the theory is to overcome the lack of unique factorisation into primes in these rings.

The course will also cover the fundamental finiteness theorems: the finiteness of the ideal class group (via Minkowski's geometric theory of numbers), and the structure (finite generation, determination of the rank, etc.) of the unit group. Additional topics which will be discussed if time permits: law of quadratic reciprocity, elementary Diophantine equations, completions (p-adic numbers), zeta-functions, distribution of primes in arithmetic progressions.

Text: None. But for those who like to see it in print: Chapter V of Zariski & Samuel, Commutative Algebra, vol.1, gives a nice, short development of the theory of Dedekind domains, with a little number theory thrown in (quadratic reciprocity). Samuel's Introduction to Algebraic Number Theory is short and elegant. Also Lang's book on number theory covers most of the material of the course (and a great deal besides).