Second Level Degree in Mathematics
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NUMBER THEORY 1 - 8 CFU
Teacher
Matteo Longo
Scheduled Period
I Year - 1 Semester | 04/10/2021 - 15/01/2022
Hours: 64 (32 esercitazione, 32 lezione)
Prerequisites
A standard Basic Algebra course; basic Linear Algebra; a basic course of Calculus; a short course in Galois Theory would be most useful; some familiarity with the theory of analytic functions of one complex variable would be useful.
Target skills and knowledge
Main points of the course will be
1) rings of algebraic integers and Dedekind domains
2) discriminants
3) quadratic and cyclotomic fields
4) decomposition of primes in a finite extension, especially if Galois
5) basic theory of valuations
6) class number of an algebraic number field
7) Finiteness of class number
8) Dirichlet's unit theorem
8) Zeta and L functions
Examination methods
Written and oral examination.
Assessment criteria
We will evaluate the level of understanding and of assimilation of the material presented in the course.
Dedication to the study and interest for the subject together with problem-solving talent will also be appreciated and evaluated.
Course contents
1. Basic algebra of commutative groups and rings.
2. Factorization of elements and ideals
3. Dedekind domains
4. Algebraic number fields. Cyclotomic and quadratic fields.
5. Rings of integers. Factorization properties.
6. Finite extensions, decomposition, ramification. Hilbert decomposition theory.
7. Frobenius automorphism, Artin map;
8. Quadratic and cyclotomic fields. Quadratic reciprocity law. Gauss sums.
9. An introduction to Class Field Theory (from Kato-Kurokawa-Saito Vol. 2, Chap. 5)
10. Minkowski Theory (finiteness of class number and the unit theorem).
11. Dirichlet series, zeta function, special values and class number formula.
Planned learning activities and teaching methods
Written and oral examination. The oral examination will consist in a discussion on the written part.
Additional notes about suggested reading
The student may find it easier to study the various topics in other textbooks or even in notes to be found online. When possible, the teacher will give suggestions on how to find the relevant material.
The notes of the course are available online on the Moodle platform.
All the textbooks proposed are freely available to the students as e-books by connecting to the Library of the Math. Department. Please ask the librarian how to connect.