2020/2021

2020/2021

DM270

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (01/10/2020 - 20/01/2021)

9

72 lesson hours

Italian

ORAL TEST

PIROLA GIAN PIETRO (titolare) - 6 ECTS
MOSCHETTI RICCARDO - 3 ECTS

Basic notion of the courses of Algebra 1 and Algebra 2:
Grous,Rings and Galoois theory.

The aim of the course is to provide an introduction to commutative algebra e di teoria dei numeri algebrici.

Commutative algebra: Modules over a (commutative) ring and operations on modules; tensor product of modules. Localization of rings and modules. Primary decomposition of ideals. Artinian and Noetherian rings and modules. Dimension theory. Integral dependence and valuations; Dedekind domains. Spectrum of a commutative ring; affine algebraic sets, Noether's normalization lemma and Hilbert's Nullstellensatz.
Affine varieties and the sheaf of regular functions. Schemes (some hints)

Number theory
Algebraic numbers, number fields, Fractiomnal ideal and divisors. Class fields. Geometric representations of numbers. Dirichlet Unity theorem. Galos theory for number Fields. Local Fields. Introduction to Minkowsky theory and to quadratic forms over number fields.

lessons

Commutative algebra: M.F. Atiyah, I.G. MacDonald: "Introduzione all'algebra commutativa", Feltrinelli, 1981. S. Bosch: "Algebraic Geometry and Commutative Algebra", Universitext, Springer, 2013. I. Kaplanski: "Commutative Rings", University of Chicago Press, 1974. H. Matsumura: "Commutative Ring Theory", Cambridge University Press, 1989.

Number Theoy Teoria dei Numeri
-Jurgen Neukirch. Algebraic Number Theory, Grundleheren der mathematischen Wissenshaffen (322) Springer (1999).
-Serge Lang, Algebraic Number Theory, Graduate texts in mathematics Spinger (1986).
-Robert Ash . A Course in algebraic number theory, Dover Books In Mathematics (2010).
-Dispense fornite dal Docente .

The exam will be a hybrid between written and oral examination.
The first part will consist of some written questions and exercises on basic concepts. Some oral questions will then be reserved for further investigation.