HIGHER ALGEBRA
Stampa
Enrollment year
2018/2019
Academic year
2018/2019
Regulations
DM270
Academic discipline
MAT/02 (ALGEBRA)
Department
DEPARTMENT OF MATHEMATICS "FELICE CASORATI"
Course
MATHEMATICS
Curriculum
PERCORSO COMUNE
Year of study
Period
1st semester (01/10/2018 - 18/01/2019)
ECTS
6
Lesson hours
48 lesson hours
Language
Italian
Activity type
ORAL TEST
Teacher
PIROLA GIAN PIETRO (titolare) - 6 ECTS
CANONACO ALBERTO - 3 ECTS
Prerequisites
The contents of the courses: Algebra 1, Algebra 2, Linear Algebra and Geometry 1.
Learning outcomes
The aim of the course is to provide an introduction to commutative algebra and to algebraic number theory.
Course contents
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.
Teaching methods
Lectures.
Reccomended or required readings
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 .
Assessment methods
Oral examination.
Further information
Oral examination.
Sustainable development goals - Agenda 2030