The course is an introduction to algebraic number theory. Starting from the definition of algebraic number, we go on studying the basic properties of rings of integers in number fields (number rings). In the final part some applications are given.
1) Marcus: Number fields
2) Stewart-Tall: Algebraic number theory and Fermat's last theorem
3) Jarvis: algebraic Number Theory
Learning Objectives
The goal of the course is to introduce the students to the basic tools needed to undertake the study of deeper subjects in number theory and Galois theory.
Prerequisites
Full programs of the courses of Algebra 1 and 2 of the first two years of Laurea Triennale in Matematica (UNIFI). It is also preferable that the student feels comfortable with the program of Algebra 3 of Laurea Triennale in Matematica (UNIFI).
Group theory, ring theory, Galois theory.
Teaching Methods
Blackboard lectures. Exercise sessions.
Type of Assessment
Oral exam.
Course program
Algebraic integers and the ring of integers of a number field (number rings). Dedekind Domains. Unique factorization of ideals in Dedekind domains. The class group. Finiteness of the class group of number rings. Ramification of rational primes and the discriminant of a number ring. Cyclotomic extensions. Kummer theory. Fermat's Last Theorem for regular primes.