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MATH-F224

Anneaux et corps commutatifs

academic year
2025-2026
2026-2027

Course teacher(s)

Joost VERCRUYSSE (Coordinator)

ECTS credits

5

Language(s) of instruction

french

Course content

In this course, we will study commutative rings and fields. The course is built around the classical Galois theorem, which explains why there is no general formula for expressing the roots of a polynomial of degree five or higher and provides a criterion for when such a formula does exist. The aim of the course is to understand this theorem, introduce the concepts and prove the results needed to understand it, and gain a good understanding of its proof.

  • rings, integral domains, fields, subrings, ring homomorphisms, the characteristic of a ring, the field of fractions of an integral domain;
  • polynomial rings, roots, irreducibility;
  • ideals, quotient rings, isomorphism theorems, prime and maximal ideals;
  • adjoining a root of a polynomial to a ring, Euclidean domains, principal ideal domains, the Chinese remainder theorem;
  • field extensions: algebraic and transcendental extensions, algebraic closures, ruler-and-compass constructions;
  • normal extensions, separable extensions, Galois extensions, the Galois correspondence and the fundamental theorem;
  • finite fields.

Objectives (and/or specific learning outcomes)

The course provides an introduction to the theory of commutative rings and fields.

By the end of this course, students will be able to understand and work with algebraic structures.

They will be able to work with these structures both through direct computations and through abstract reasoning.

Prerequisites and Corequisites

Required and Corequired knowledge and skills

Good knowledge of basic notions in algebra (MathF121), linear algebra (MathF122) and group theory (MathF223).

Required and corequired courses

Courses requiring this course

Teaching methods and learning activities

Lectures and guided exercise sessions.

References, bibliography, and recommended reading

Lecture notes will be available via UV.
Principal reference:
S. Lang, Algebra. Revised third edition, Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. (ISBN: 0-387-95385-X)
Additional references:
M. Artin, Algebra, Prentice Hall, London, 1991. (ISBN: 0-13-004763-5)
P.M. Cohn, Algebra, Vol. 1, John Wiley & Sons, London, 1974. (ISBN: 0-471- 16431-3)
N. Jacobson, Basic algebra I. Second edition, W. H. Freeman and Company, New York, 1985. (ISBN: 0-7167-1480-9)


 

Course notes

  • Syllabus
  • Université virtuelle

Contribution to the teaching profile

1. Acquiring and applying knowledge

1.1. Acquire a sound understanding of fundamental mathematical concepts.

1.2. Master the basic concepts of algebra, analysis and geometry.

1.3. Analyse, synthesise and connect knowledge across the different branches of mathematics.

1.4. Master the principles of logical reasoning and be able to base rigorous arguments on them.

1.6. Identify the underlying mathematical framework of a given problem.

1.8. Learn to develop one's knowledge, in particular by seeking out and critically evaluating information.

2. Understanding and practising the specific approach of scientific inquiry

2.1. Understand the criteria of rigour, argumentation and proof techniques.

2.2. Understand how a concept emerges from observations and examples.

2.3. Understand the process of abstraction and its role in the development of a theory.

2.5. Understand the sometimes simplifying role of the process of generalising a theory.

2.6. Understand the value of unifying existing theories.

2.7. Identify questions that arise within a theory.

2.8. Explore the consequences of a mathematical result.

3. Communication

3.1. Design and write a mathematical result or theory rigorously.

3.3. Use clear and rigorous language adapted to the target audience.

4. Ethics and relationship with society

4.1. Take responsibility for one's statements.

4.3. Learn to critically assess the validity of an argument.

4.4. Prohibit all forms of plagiarism.

Other information

Contacts

Joost VERCRUYSSE <joost.vercruysse@ulb.be>

Campus

Plaine

Evaluation

Method(s) of evaluation

  • written examination
  • Oral examination

written examination

Oral examination

Written examination, followed a few days later by an oral examination during which the questions and answers from the written examination are discussed again.

Mark calculation method (including weighting of intermediary marks)

Mark on 20 based on both written and oral examination.

Language(s) of evaluation

  • french

Programmes