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 ideal-membership or the radical-membership, 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

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 !