Instructor: Dr. András BÍRÓ

Text: Chapters 1 and 2 of J. Neukirch: Algebraic Number Theory, Springer, 1999

Reference book: J. Neukirch: Algebraic Number Theory, Springer, 1999

Prerequisite: elementary number theory, basic theorems of linear algebra, basic notions of abstract algebra (fields, rings, modules)

Course description:the aim of the course is to present the basic properties of algebraic number fields (i.e finite extensions of the field of rational numbers)

Topics:

integral closure of a subring, existence of an integral basis

Dedekind domains, unique factorization of prime ideals

Dedekind domains, unique factorization of prime ideals

structure of the units of an algebraic number field: Dirichlet`s Unit Theorem

decomposition of a prime ideal in an extension

cyclotomic fields

p-adic numbers, local fields