
Course unit
NUMBER THEORY 1
SCP4063857, A.A. 2015/16
Information concerning the students who enrolled in A.Y. 2015/16
Mutuating
Course unit code 
Course unit name 
Teacher in charge 
Degree course code 
SCP4063857 
NUMBER THEORY 1 
FRANCESCO BALDASSARRI 
SC1172 
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/03 
Geometry 
8.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
5 Teoria dei Numeri 1  a.a. 2018/2019 
01/10/2018 
30/09/2019 
BALDASSARRI
FRANCESCO
(Presidente)
LONGO
MATTEO
(Membro Effettivo)
CAILOTTO
MAURIZIO
(Supplente)
CANDILERA
MAURIZIO
(Supplente)
KLOOSTERMAN
REMKE NANNE
(Supplente)

4 Teoria dei Numeri 1  2017/2018 
01/10/2017 
30/09/2018 
BALDASSARRI
FRANCESCO
(Presidente)
LONGO
MATTEO
(Membro Effettivo)
CAILOTTO
MAURIZIO
(Supplente)
CANDILERA
MAURIZIO
(Supplente)
KLOOSTERMAN
REMKE NANNE
(Supplente)

3 Teoria dei Numeri 1  2016/2017 
01/10/2016 
30/11/2017 
BALDASSARRI
FRANCESCO
(Presidente)
LONGO
MATTEO
(Membro Effettivo)
CAILOTTO
MAURIZIO
(Supplente)
CANDILERA
MAURIZIO
(Supplente)
KLOOSTERMAN
REMKE NANNE
(Supplente)

2 Teoria dei Numeri 1  a.a. 2015/2016 
01/10/2015 
30/09/2016 
BALDASSARRI
FRANCESCO
(Presidente)
BERTAPELLE
ALESSANDRA
(Membro Effettivo)
CAILOTTO
MAURIZIO
(Supplente)
CANDILERA
MAURIZIO
(Supplente)
ESPOSITO
FRANCESCO
(Supplente)

Prerequisites:

A standard Basic Algebra course; the experience of a short course in Galois Theory would be most useful; Linear Algebra; Notions of Calculus; some familiarity with complex functions of one variable might be useful. 
Target skills and knowledge:

Algebraic number fields and rings. Explicit determination of the ring of integers of quadratic, cyclotomic (and of some cubic) fields. The theory of discriminant and of ramification. Decomposition of primes. Galois and Hilbert theories.Quadratic Reciprocity Law. Minkowski theory. Determination of the class and the unit group in simple cases. Introduction to Class Field Theory. 
Examination methods:

Three written partials will be proposed during the course. They are supposed to check the stepbystep understanding of the course by the students. A final allinclusive exam will be proposed for those who have not passed the partials or are not satisfied with the grades.
Each student will be invited 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:

Dedication to the study and interest for the subject together with problemsolving talent will 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 theory.
7. Frobenius automorphism, Artin map;
8. Quadratic and cyclotomic fields. Quadratic reciprocity law. Gauss sums.
9. An introduction to Class Field Theory (from KatoKurokawaSaito Vol. 2, Chap. 5)
10. Minkowski Theory (finiteness of class number and the unit theorem).
11. Hilbert symbols (from KatoKurokawaSaito, Vol. 1 Cap. 2).
12. Dirichlet series, zeta function, special values and class number formula (from KatoKurokawaSaito, Vol. 1).
The whole material is to be found in the single textbook: Daniel A. Marcus "Number Theory", SpringerVerlag. Our essential program consists of Chapters 1 to 5, with those exercises which are used in the body of the textbook. The complexanalytic proofs in Chapter 5 will not be required.
We recommend, for cultural reasons, reading through the two volumes of KatoKurokawaSaito, possibly without studying proofs. 
Planned learning activities and teaching methods:

The written partials are meant to be a test of the stepbystep understanding of the course by the students. Very often the exercises proposed will be taken from (previously indicated) sections of the textbook. This should encourage the students to try in advance the exercisies of the book.
Each student will be invited 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 final oral examination consists in a lecture to be held by the student in a separate session on a higherlevel topic. 
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.

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.


