Algebraic Number Theory
A.Y. 2020/2021
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
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.
-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.
-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
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)