Algebraic Number Theory | Università degli Studi di Milano Statale

A.Y. 2024/2025

Max ECTS

Overall hours

SSD

MAT/02

Language

Italian

Included in the following degree programmes

Mathematics (Classe LM-40)-Enrolled from 2012/13

Learning objectives

(first part) The course provides standard results in algebraic number theory, hence introduce L-functions and their arithmetic relevance.

Expected learning outcomes

(first part) 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: First semester

Lessons timetable

Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi

Exams calendar

Single course

This course can be attended as a single course.

Take a single course

Course syllabus and organization

Algebraic number theory (activated first part)

Responsible

Seveso Marco Adamo

Lesson period

First semester

Syllabus
(active tab)
Course structure

Prerequisites for admission

Basic knowledge of algebra (Algebra 1-4) and analysis (Analisi Matematica
1-4 and Analisi complessa).

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 ramication).
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

-J. W. S. Cassels and A. Frohlich, "Algebraic number theory", Academic Press Inc. (London) LDT, 1967.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields I and II", combined 2nd edition, Springer, 2012.
-D. A. Marcus, "Number fields", Springer, 2018 (2nd edition).
-R. Schoof, "Algebraic number theory". Notes avalable at https://www.mat.uniroma2.it/~eal/moonen.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

-J. W. S. Cassels e A. Frohlich, "Algebraic number theory", Academic Press Inc. (London) LDT, 1967.
-S. Lang, "Algebraic number theory", Springer, 1994.
-S. Lang, "Cyclotomic fields 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

Lessons: 42 hours

Professor: Seveso Marco Adamo

Number Theory mod/2

MAT/02 - ALGEBRA - University credits: 3

Lessons: 21 hours

Professor(s)

Seveso Marco Adamo

Web site

Reception:

On appointment

Via Cesare Saldini 50

We use a selection of our own and third-party cookies on the pages of this website: Essential cookies, which are required in order to use the website; functional cookies, which provide better easy of use when using the website; performance cookies, which we use to generate aggregated data on website use and statistics; and marketing cookies, which are used to display relevant content and advertising. If you choose "ACCEPT ALL", you consent to the use of all cookies. You can accept and reject individual cookie types and revoke your consent for the future at any time at "Settings".