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.

TEACHING METHODS

Traditional method: lectures in presence.

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.
• Ramification in quadratic fields.
• 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.
• Quadratic reciprocity law.
• 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.

RECOMMENDED READING/BIBLIOGRAPHY

• 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

LESSONS

Class schedule

The timetable for this course is available here: EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam.

ASSESSMENT METHODS

The exam will be mainly based on the topics explained during the lectures. The goal of the exam will be to check, on the one hand, the knowledge of the student of the notions introduced and the results proved in the course and, on the other hand, his/her ability to solve exercises analogous to those proposed in the exercise sheets available on Aulaweb.

Exam schedule