CODE  90694 

ACADEMIC YEAR  2024/2025 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/02 
LANGUAGE  Italian 
TEACHING LOCATION 

SEMESTER  1° Semester 
TEACHING MATERIALS  AULAWEB 
OVERVIEW
Algebra 3 covers the basics of commutative algebra and the associated algorithmic and computational aspects. In particular, the notion of Noetherian rings and modules and of algebra are presented and subsequently the Groebner bases are introduced. Groebner bases are then applied to get computational resolutions of problems related to polynomial objects.
AIMS AND CONTENT
LEARNING OUTCOMES
The aim of the course is to provide students with an introduction to commutative algebra and related computational issues. The keywords are Neotherian rings, modules over them, Groebner bases and solutions of systems of polynomial equations.
AIMS AND LEARNING OUTCOMES
Algebra 3 aims to provide the basics of:
1) Noetherian rings and modules
2) Algebras
3) Groebner bases and their use for symbolic computations
4) Algorithms related to the computation and use of Groebner bases.
The expected learning outcomes are:
1) At the end of Algebra 3 one is able to recognize a ring or a Noetherian module and to illustrate its salient properties also through the reproduction of the main proofs associated with them.
2) At the end of Algebra 3 one is able to solve exercises relating to ideals and modules with a particular structure such as, for example, monomial ideals.
3) At the end of Algebra 3, one is able to illustrate how the use of Groebner bases allows to algorithmically solve problems such as the idealmembership or the radicalmembership, the calculation of the intersection of ideals and their syzygies, the elimination of variables and the computation of kernels of homomorphisms of algebras.
4) At the end of Algebra 1 one is able to decide algorithmically whether a system of polynomial equations has solutions and to describe them as explicitly as possible.
5) At the end of Algebra 1 one is able to program a symbolic computation systems such as CoCoA5 to solve concrete computational problems.
PREREQUISITES
Knowledge of basic algebraic structures: vector spaces, groups and rings.
TEACHING METHODS
Blackboard lectures and computer exercises with the use of symbolic calculation programs
SYLLABUS/CONTENT
Rings, ideals and modules.
Noetherian rings and the Hilbert basis theorem.
Polynomials in several variables with coefficients in a field.
Monomial ideals.
Algebras
Gröbner bases, ordering and division with remainder.
Buchberger algorithm.
Membership of a polynomial in an ideal its radical. Intersection of ideals. Elimination. Calculation of kernels of maps between algebras and between modules.
Solution of systems of polynomial equations.
Computational and implementation aspects of Groebner bases.
RECOMMENDED READING/BIBLIOGRAPHY
Computational Commutative Algebra, Kreuzer, Robbiano, Springer, 2004.
TEACHERS AND EXAM BOARD
Ricevimento: Office hours will be fixed at the beginning of the semester and comunicated via alulaweb.
Ricevimento: Reception hours: before and after lessons, or upon request by email/Teams.
LESSONS
LESSONS START
The class will start according to the academic calendar.
Class schedule
The timetable for this course is available here: Portale EasyAcademy
EXAMS
EXAM DESCRIPTION
Oral exams and computational lab exercises
ASSESSMENT METHODS
The oral exam consists of
1) the discussion of a basic notion among those presented in the first part of the course illustrated through relevant examples and proofs
2) the discussion and illustration of one of the basic computational problems and its algorithmic solution through the use of Groebner bases with relative proofs and algorithmic procedures.
The laboratory part consists in the writing of symbolic calculation programs using CoCoA5 for the solution of concrete problems proposed.
FURTHER INFORMATION
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the office Settore Servizi di supporto alla disabilità e agli studenti con DSA and the teachers at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools. Good afternoon !