Enroll Course: https://www.coursera.org/learn/theorie-de-galois
Overview
The theory of Galois is a cornerstone of modern algebra, linking polynomial roots and group theory in a way that fundamentally alters our understanding of equations. The Coursera course, Introduction à la théorie de Galois, offers an in-depth exploration of these concepts, starting from the classical non-solvability criteria of polynomial equations to advanced methods for calculating Galois groups through reduction modulo prime numbers.
In this course, students will delve into the symmetry of polynomial roots and learn how to express these roots based on the coefficients of the polynomial. Evariste Galois’ pioneering ideas will be emphasized throughout this academic journey.
Syllabus Highlights
The course is structured into several key modules:
- Introduction: Students will begin with a description of the problem concerning polynomial roots, gaining foundational insights.
- Extensions de corps: This section dives into algebraic closure, primitive element lemmas, and related topics.
- Polynôme minimal: Here, students explore conjugate elements, leading to a deeper understanding of minimal polynomials.
- Corps fini: A focus on finite fields, including Frobenius, automorphisms, and extensions.
- Théorie des groupes I & II: Students will learn fundamental results regarding groups, including Lagrange’s theorem and solvability criteria for symmetric groups.
- Correspondance de Galois: This crucial section covers Galois groups and their correspondence.
- Cyclotomie I: General cyclotomic extensions and Kummer’s theory are introduced.
- Théorèmes de résolubilité de Galois: Students will investigate criteria for solvability and Galois’ theorem for degree p polynomials.
- Réduction mod p: Learn how to apply Galois groups to polynomials with integer coefficients via modular reduction.
- Compléments: Cyclotomic fields over Q and additional applications will round out the course content.
Why You Should Take This Course
This course is a fantastic opportunity for anyone looking to deepen their understanding of algebra and polynomial theory. It’s suitable for advanced undergraduates or graduate students in mathematics, as well as professionals seeking to refresh or expand their knowledge.
Engaging lectures, comprehensive materials, and practical examples make this course highly effective for mastering a complex subject area. With the inclusion of group theory and its applications, students gain not only theoretical knowledge but also practical skills that are invaluable in various scientific and engineering fields.
Final Thoughts
If you’re intrigued by the mathematical structure behind polynomial equations and looking to explore one of the most fascinating topics in algebra, Introduction à la théorie de Galois is the course for you. Its well-organized syllabus, expert instructors, and rigorous content will provide you with a solid foundation in Galois theory.
Don’t miss out on this opportunity—enroll today and unlock the secrets of polynomial roots!
Enroll Course: https://www.coursera.org/learn/theorie-de-galois