CODE 38752 2023/2024 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA MAT/02 Italian GENOVA 1° Semester AULAWEB

## OVERVIEW

The course offers an introduction to Algebraic Number Theory.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The purpose of the course is to introduce the basic algebraic notions, and the corresponding techniques, that are used in the study of the arithmetic of number fields and, more generally, of Dedekind domains. The course provides the necessary algebraic background to deal with more sophisticated questions in Number Theory, Arithmetic Geometry and related topics.

### AIMS AND LEARNING OUTCOMES

At the end of the course, students will have a good knowledge of basic notions in Algebraic Number Theory, such as unique factorization of ideals in Dedekind domains, ramification of prime ideals in (Galois) extensions of number fields, ideal class group of a Dedekind domain, p-adic numbers.

### PREREQUISITES

All courses (in particular, the algebra courses) from the first two years of Laurea in Matematica.

### SYLLABUS/CONTENT

• Review and background of basic algebraic results.
• Integral dependence; integrally closed domains.
• Generalities on field extensions.
• Primitive element theorem and its consequences.
• Norm and trace of an element.
• Fractional ideals of an integral domain.
• Dedekind domains.
• Unique factorization of ideals in a Dedekind domain.
• Class group of a Dedekind domain.
• Class group and class number of a number field.
• Hermite-Minkowski theorem, Hermite theorem, Dirichlet's unit theorem.
• Ramification of prime ideals.
• Ramification and discriminant.
• The fundamental theorem of Galois theory in characteristic 0.
• Hilbert's ramification theory, decomposition group and inertia group.
• Frobenius automorphism.
• Cyclotomic fields: rings of integers and discriminants.
• p-adic numbers: definitions and basic properties.
• Hensel's lemma and some of its applications.
• Local-global principle: statement and some examples.
• Hasse-Minkowski theorem: statement.

• S. Lang, Algebraic number theory, second edition, Springer, 1994.
• D. A. Marcus, Number fields, second edition, Springer, 2018.
• J. Neukirch, Algebraic number theory, Springer, 1999.
• P. Samuel, Algebraic theory of numbers, Dover, 2008.

## TEACHERS AND EXAM BOARD

### Exam Board

STEFANO VIGNI (President)

SANDRO BETTIN

FRANCESCO VENEZIANO (Substitute)

## LESSONS

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### Exam schedule

Data Ora Luogo Degree type Note
08/01/2024 09:00 GENOVA Esame su appuntamento
03/06/2024 09:00 GENOVA Esame su appuntamento