Algebraic Number Theory

A.Y. 2021/2022
9
Max ECTS
73
Overall hours
SSD
MAT/02
Language
Italian
Learning objectives
The course provides standard results in algebraic number theory, hence introduce L-functions and their arithmetic relevance.
Expected learning outcomes
Learning the basic results in Algebraic Number Theory. Ability of computing the class groups and the group of units of a number field. Acquire familiarity with L-functions and other more advanced topics.
Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Prerequisites for admission
Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica 1-4).
Assessment methods and Criteria
The exam consists of an oral examination.
Number Theory (first part)
Course syllabus
First properties of a number field, review of some of the arguments of the Algebra 3 course (Dedekind rings, factorization of ideals and ramification). Theorem of Minkowski. Theorem of Hermite. Theorem of Dirichlet and regulator of a number field. Dedekind zeta function. Class number formula. Complex L functions, special value formulas and relationship with cyclotomic units.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-D. A. Marcus, "Number fields", Springer, 2018 (2nd edition).
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fileds I and II", combined 2nd edition, Springer, 2012.
-R. Schoof, "Algebraic number theory". Notes avalable at https://www.mat.uniroma2.it/~schoof/TNT2.pdf.
-J.-P. Serre, "Local fields", Springer.
-L. C. Washington, "Introduction to cyclotomic fields", 2nd edition, Springer, 1997.
Number Theory mod/2
Course syllabus
Other arithmetic applications of L-functions.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fileds I and II", combined 2nd edition, Springer, 2012.
-J.-P. Serre, "Local fields", Springer.
-L. C. Washington, "Introduction to cyclotomic fields", 2nd edition, Springer, 1997.
Number Theory (first part)
MAT/02 - ALGEBRA - University credits: 6
Practicals: 24 hours
Lessons: 28 hours
Professor: Seveso Marco Adamo
Number Theory mod/2
MAT/02 - ALGEBRA - University credits: 3
Lessons: 21 hours
Professor: Venerucci Rodolfo
Professor(s)
Reception:
On appointment
Via Cesare Saldini 50