Algebraic Number Theory

A.Y. 2020/2021
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
Prerequisites for admission
Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica 1-4).
Assessment methods and Criteria
L'esame consiste di una discussione orale.
Number Theory (first part)
Course syllabus
First properties of a number field: norm, trace, discriminant and ring of integers (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 ζ function. Class number formula.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-Note di R. Schoof, "Algebraic number theory ". Avalable at https://www.mat.uniroma2.it/~schoof/TNT2.pdf.
-J.-P. Serre, "Local fields", Springer.
Number Theory mod/2
Course syllabus
L-functions, special values and arithmetic applications. p-adic L-functions and p-adic variants of the class number formula.
Teaching methods
Whole class teaching of theory and excercises.
Teaching Resources
-S. Lang, "Algebraic number theory", Springer.
-H. Hida, "Elementary theory of L-functions and Eisenstein series", London Mathematical Society Students Texts 26.
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