Syllabus
Algebraic integers
Dedekind domains
Prime factorization, ramification and splitting
Geometry of numbers, Minkowski bound
Finiteness of class group
Dirichlet unit theorem, regulator
Cyclotomic fields, Diophantine equations p-adic numbers
Ramification and inertia group, Frobenius element
Annotation
Algebraic number theory studies the structure of number fields and forms the basis for most of advanced areas of number theory. In the course we will develop its main tools that are connected to algebraic integers, prime ideals, ideal class group, unit group, and subgroups of the Galois group, including basics of p-adic numbers and applications to Diophantine equations.