OVERVIEW

Algebra 3 deals with the basic concepts of commutative algebra and the algorithmic and computational aspects associated with them. In particular, it introduces the notions of Noetherian rings and modules, as well as algebras, followed by an introduction to Gröbner bases, which are applied to the computational resolution of problems related to polynomial objects, both theoretically and in the laboratory.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide students with a solid understanding of the fundamental concepts of commutative algebra and the related computational aspects. In particular, the notions of Noetherian ring, module, Groebner basis are discussed and the some methods to solve of systems of polynomial equations are presented.

AIMS AND LEARNING OUTCOMES

Algebra 3 aims to provide students with fundamental knowledge in the following topics:

1. Noetherian rings and modules

2. Algebras

3. Gröbner bases and their use in symbolic computation

4. Algorithms related to the computation and application of Gröbner bases

Expected learning outcomes:

1. By the end of Algebra 3, students will be able to recognise a Noetherian ring or module and describe its main properties, including reproducing the principal proofs associated with them.

2. By the end of Algebra 3, students will be able to solve exercises concerning ideals, algebras, and modules with specific structures, such as monomial ideals or modules over PIDs.

3. By the end of Algebra 3, students will be able to explain how Gröbner bases can be used to algorithmically solve problems such as testing whether a polynomial belongs to an ideal or its radical, computing intersections of ideals and their syzygies, eliminating variables, and computing kernels of algebra homomorphisms.

4. By the end of Algebra 3, students will be able to determine algorithmically whether a system of polynomial equations has solutions, and describe these solutions explicitly whenever possible.

5. By the end of Algebra 3, students will be able to use symbolic computation systems such as CoCoA5 to solve concrete computational problems.

PREREQUISITES

The prerequisites include knowledge of basic algebraic structures such as vector spaces, groups, and rings.

TEACHING METHODS

1. Lectures delivered by the instructors, during which theoretical content will be presented, applied to concrete examples, and illustrated through problem-solving exercises.

2. Laboratory sessions in computational algebra, where instructors will demonstrate the practical implementation of the algorithms studied and explained during the theoretical lectures.

Students with certified Specific Learning Disorders (SLD), disabilities, or other educational needs are encouraged to contact the instructors and the School/Department’s disability coordinator at the beginning of the course to arrange any necessary teaching adaptations. These will respect the learning objectives while taking individual learning needs into account.

Students who are unable to attend the lectures are invited to contact the instructors in order to arrange dedicated meetings and receive specific course materials.

SYLLABUS/CONTENT

Rings, ideals, and modules. Noetherian rings and modules. Algebras over a ring. Hilbert’s Basis Theorem. The universal property of the polynomial ring in several variables over a ring. Monomial ideals. The structure theorem for modules over principal ideal domains (PIDs).

Orderings, Gröbner bases, and division with remainder. S-pairs and syzygies. Buchberger’s criterion and algorithm. Compatibility of structural properties with Buchberger’s algorithm. Computational and implementation aspects of Gröbner bases. Algorithmic solutions to the following problems, addressed both theoretically and through computational laboratory work:

1. Testing whether a polynomial belongs to an ideal

2. Testing whether a polynomial belongs to the radical of an ideal

3. Computation of syzygies

4. Intersection of ideals

5. Polynomial elimination

6. Computation of kernels of maps between algebras and between modules

7. Solving systems of polynomial equations

RECOMMENDED READING/BIBLIOGRAPHY

• Computational Commutative Algebra 1
  Authors: Martin Kreuzer, Lorenzo Robbiano
  Springer, 2000.

• Informal Notes on Computational Algebra
  Author: Aldo Conca, 2020 (available on AulaWeb)

• Informal Notes on Commutative Algebra
  Author: Matteo Varbaro, 2022 (available on AulaWeb)

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

22 September 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral examination and a computational laboratory test.

• The oral examination is graded with a score T in the range [0,29] and is considered passed with a score of 18 or higher.

• The laboratory test is graded with a score L in the range [0,1.5].

The final grade is the sum T + L.

The two components may be taken on different dates and during separate exam sessions.

ASSESSMENT METHODS

The oral examination consists of:

1. A discussion of a fundamental concept from the first part of the course, illustrated with relevant examples and proofs.

2. A discussion and explanation of one of the basic computational problems and its algorithmic solution using Gröbner bases, including the relevant proofs, algorithmic procedures, and significant examples.

The laboratory test consists of writing symbolic computation programs using CoCoA5 to solve two concrete problems.

FURTHER INFORMATION

Attendance is strongly recommended, as both the lectures and laboratory sessions are essential for understanding the course content. The material covered combines theoretical and practical aspects of algebra and is often guided by heuristic considerations not typically found in textbooks.

Students with certified Specific Learning Disorders (SLD), disabilities, or other special educational needs are advised to contact the Support Services for Students with Disabilities and SLD, as well as the course instructors, to obtain information about available learning and assessment arrangements, including appropriate compensatory tools.