Second Level Degree in Mathematics
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NUMBER THEORY 1 - 8 CFU
Teacher
Francesco Baldassarri
Scheduled Period
I Year - 2 Semester | 02/03/2020 - 12/06/2020
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) class number of an algebraic number field
6) Finiteness of class number
7) Dirichlet's unit theorem
8) Zeta and L functions
Examination methods
We will propose the preparation of 1 or 2 written reports during the course. These are supposed to check the step-by-step understanding of the topics presented and the interest of the students in the subject. The exam will be concluded by a final report on a topic chosen by the teacher that the student will prepare individually at home.
Students will be offered to present one topic agreed with the teacher in a 45 minutes lecture during the course. A final oral examination is reserved for those who aim at top grades.
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.
The whole material is to be found in the single textbook: Daniel A. Marcus "Number Theory", Springer-Verlag. The essential part of the program consists of Chapters 1 to 5, with those exercises which are used in the body of the textbook.
Chapters 6 and 7 are required to get a higher grade. The lengthy real-analytic proofs in Chapters 5/6/7 are not essential. A good understanding of the complex-analytic strategy is necessary.
We recommend, for cultural reasons, reading through the two volumes of Kato-Kurokawa-Saito, possibly without studying proofs.
Planned learning activities and teaching methods
The 1 or 2 reports proposed during the semester are meant to be a test of the step-by-step understanding of the course by the students. Very often the topics proposed will be taken from exercise sections of the textbook. This should encourage the students to try by themselves the exercisies of the book.
Every student will be offered to present one topic agreed with the teacher in a 45 minutes lecture during the course. This is supposed to show the expository ability of the student.
The possible final oral examination consists in a presentation to be held by the student in a separate session on a topic indicated by the teacher a couple of hours in advance. The student is supposed to use those hours to refresh his preparation.
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.