*Note from June 29, 2020*

Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.

Cours LMAT1131.

Topics discussed in the course : solution of systems of algebraic equations. Arithmetic of polynomial rings and elimination theory. Structure of modules over a principal ideal domain, and application to the classification of linear operators on finite-dimensional vector spaces.

*The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.*

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1 | Contribution of the course to learning outcomes in the Bachelor in Mathematics programme. By the end of this activity, students will have made progress in: - Recognising and understanding a basic foundation of mathematics. -- Choosing and using the basic tools of calculation to solve mathematical problems. -- Recognising the fundamental concepts of important current mathematical theories. -- Establishing the main connections between these theories, analysing them and explaining them through the use of examples. - Identifying, by use of the abstract and experimental approach specific to the exact sciences, the unifying features of different situations and experiments in mathematics or in closely related fields (probability and statistics, physics, computing). - Showing evidence of abstract thinking and of a critical spirit : -- Arguing within the context of the axiomatic method, recognising the key arguments and the structure of a proof, constructing and drawing up a proof independently. -- Evaluating the rigour of a mathematical or logical argument and identifying any possible flaws in it. -- Distinguishing between the intuition and the validity of a result and the different levels of rigorous understanding of this same result.
Learning outcomes specific to the course. By the end of this activity, students will be able to: - Factor multivariate polynomials into irreducible factors. - Analyse systems of algebraic equations to determine whether they admit solutions and to represent them geometrically. - Find equations with a given parametrised set of solutions. - Analyse the structure of modules over a principal ideal domain. - Reduce linear operators over a finite-dimensional vector space to canonical forms. |

*The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.*

The following topics are discussed :

- Commutative rings and ideals

- Local rings

- Modules, exact sequences, tensor products

- Noethérian rings.

- Polynomials and affine algebraic varieties.

- Groebner bases of ideals in polynomial rings.

- Unique factorisation, resultants, and elimination in polynomial rings.

- Existence of solutions for systems of algebraic equations (Hilbert's Nullstellensatz).

- Localisation

- Structure of modules of finite type over principal ideal domains