First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
NUMBER THEORY 1
SCP4063857, A.A. 2015/16

Information concerning the students who enrolled in A.Y. 2015/16

Information on the course unit
Degree course Second cycle degree in
MATHEMATICS
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2015/16
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Degree course track ALGANT [001PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination NUMBER THEORY 1
Website of the academic structure http://matematica.scienze.unipd.it/2015/laurea_magistrale
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Branch PADOVA
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Lecturers
Teacher in charge FRANCESCO BALDASSARRI MAT/03

Mutuating
Course unit code Course unit name Teacher in charge Degree course code
SCP4063857 NUMBER THEORY 1 FRANCESCO BALDASSARRI SC1172

ECTS: details
Type Scientific-Disciplinary 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

Calendar
Start of activities 01/10/2015
End of activities 28/01/2016
Show course schedule 2019/20 Reg.2011 course timetable

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)

Syllabus
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 step-by-step understanding of the course by the students. A final all-inclusive 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 problem-solving 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 Kato-Kurokawa-Saito Vol. 2, Chap. 5)
10. Minkowski Theory (finiteness of class number and the unit theorem).
11. Hilbert symbols (from Kato-Kurokawa-Saito, Vol. 1 Cap. 2).
12. Dirichlet series, zeta function, special values and class number formula (from Kato-Kurokawa-Saito, Vol. 1).

The whole material is to be found in the single textbook: Daniel A. Marcus "Number Theory", Springer-Verlag. Our essential program consists of Chapters 1 to 5, with those exercises which are used in the body of the textbook. The complex-analytic proofs in Chapter 5 will not be required.
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 written partials are meant to be a test of the step-by-step 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 higher-level 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. 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. Cerca nel catalogo