2018/2019

2018/2019

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (01/10/2018 - 18/01/2019)

6

48 lesson hours

Italian

ORAL TEST

PIROLA GIAN PIETRO (titolare) - 6 ECTS
CANONACO ALBERTO - 3 ECTS

The contents of the courses: Algebra 1, Algebra 2, Linear Algebra and Geometry 1.

The aim of the course is to provide an introduction to commutative algebra and to algebraic number theory.

Commutative algebra
Modules over a (commutative) ring and operations on modules; tensor product of modules. Localization of rings and modules. Artinian and Noetherian rings and modules; Krull dimension of a ring. Integral dependence. Spectrum of a ring; affine algebraic sets, Noether's normalization lemma and Hilbert's Nullstellensatz.

Number Theory
Algebraic numbers and algebraic integers. Number Fields.
Factorization in Dedekind domains and divisors. Ideal and class group.
Geometric representation of algebraic number. Dirichlet unity theorem. Galois theory for number fields.
Valuations Local fields. Introduction to the Minkowsky theory and Riemann Roch theorem.

Lectures.

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 Thory
-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 .

Oral examination.

Oral examination.