First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP4063857, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2017/18
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination NUMBER THEORY 1
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English


Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/02 Algebra 2.0
Core courses MAT/03 Geometry 3.0
Core courses MAT/05 Mathematical Analysis 3.0

Mode of delivery (when and how)
Period First semester
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
Hours of
Individual study
Practice 4.0 32 68.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 02/10/2017
End of activities 19/01/2018

Prerequisites: A standard Basic Algebra course; a short course in Galois Theory would be most useful; basic Linear Algebra; a basic course of Calculus; some familiarity with the theory of analytic functions of one complex variable would be useful.
Target skills and knowledge: Algebraic number fields and rings. Rings of algebraic integers. Explicit determination of the ring of integers of quadratic, cyclotomic (and of some cubic) fields. Elementary theory of discriminants and of ramification. Decomposition of primes in Dedekind rings. Field extensions and decomposition of primes in an extension. Galois extensions and Hilbert theory. Decomposition and inertia groups.
Different ideal and higher ramification groups. Abelian extensions, unramified extensions. Frobenius. Explicit determination of decomposition and inertia subgroups for any rational prime in cyclotomic fields. Quadratic subfields of cyclotomic fields. Quadratic reciprocity law. Characters of finite abelian groups. Gauss sums. Minkowski theory. Finiteness of the class group and Dirichlet Units Theorem. Regulator. Determination of the class and the unit group in simple cases. Units of real quadratic fields and Pell equation. Distribution of ideals in a ring of algebraic integers. Determination of the constant in the asymptotic formula. Analytic theory of Dirichlet series. Dedekind zeta function. Dirichlet L functions. Polar and Dirichlet densities. The Class Number Formula. Calculation of L-series at 1 and Gauss sums. The quadratic case.
Introduction to Class Field Theory.
Examination methods: We will propose the preparation of 2 or 3 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. A final all-inclusive exam will be proposed for those who have not presented satisfactory reports during the year as well as to those who are not satisfied with the mark obtained.

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 unit 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 2 or 3 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.
Textbooks (and optional supplementary readings)
  • Daniel A. Marcus, Number Fields. --: Springer Universitext, 1977. Cerca nel catalogo
  • Kazuya Kato, Nobushige Kurokawa, Takeshi Saito, Number Theory 1 (Fermat's Dream) and Number Theory 2 (Introduction to Class Field Theory). --: Translations of Math. Monographs Vol. 186 and 240 American Mathematical Society, 2011.