Preparation for the Elective

This course prepares students for the modeling exam in Option C of the external mathematics certification exam. This exam, which is based on the analysis of texts, focuses on the connections between algebra and symbolic computation. The symbolic computation software SageMaths is used in this course.

The topics covered are those of Option C’s curriculum: representation and algorithmic manipulation of common mathematical objects in algebra and formal computation (integers, floating-point numbers, integers modulo n, polynomials, matrices); machine-imposed limitations (space and time optimization, the concept of algorithmic complexity), and application areas for these theories (error-correcting codes, cryptography, information processing and data compression, geometry, etc.).

Classical algorithms (fast exponentiation, extended Euclidean algorithm, Hörner's scheme, Gauss's algorithm, modular methods, primality tests, etc.) are introduced in class and then explored on the computer using SageMath software. This is also an opportunity to become familiar with this software.

Each student is required to present one or more oral lessons—depending on the requirements of the agrégation—based on texts from previous years.

Prepare students for Option C of the agrégation, which requires:

• apply the mathematical concepts and tools in the curriculum with precision and rigor;
• distinguish between exact and approximate representations of mathematical objects;
• estimate the cost and limitations of simple algorithms: complexity, accuracy;
• analyze the relevance of the models.
• know how to implement a formal computation algorithm.

The main prerequisites are the concepts and techniques covered in the algebra curriculum for the third year of the Bachelor’s degree in mathematics: common algebraic structures (group, ring, field, vector space, module over a principal ring, algebra); elementary arithmetic and the Euclidean algorithm (extended); polynomials in one and multiple variables; finite fields and field extension theory; matrices and the Gauss algorithm

Recommended prerequisites: A strong background in algebra and formal calculus is beneficial, but proficiency beyond the necessary prerequisites mentioned above is not required.

Comprehensive Continuous Monitoring

See the EU description and detailed syllabus for the modeling exam in Option C of the external mathematics certification exam

Hourly volumes:

            CM: 22

            TD: 22

            TP: 0

            Land: 0