Introduction à la théorie de Galois

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Introduction

# Course Review: Introduction à la théorie de Galois *Platform: Coursera* ### Overview "Introduction à la théorie de Galois" is a profound course designed to delve into the essential elements of Galois theory—a critical pillar in abstract algebra that touches on the solvability of polynomial equations through the innovative lens of symmetry and group theory. Facilitated in French, this course covers an array of topics, starting from the classical criteria for the unsolvability of polynomial equations up to more advanced techniques for calculating Galois groups via reductions modulo a prime number. The course thoughtfully unravels the complex world of polynomials, offering insights into the roots of these equations and the relationships between these roots as governed by their coefficients. Evariste Galois' groundbreaking work allows students to explore how polynomial roots exhibit symmetrical properties, leading into the captivating study of permutation groups associated with these roots. ### Syllabus Breakdown The course is structured into several key modules that progressively build on one another: 1. **Introduction**: This module serves as a warm-up, introducing students to the problems inherent in single-variable polynomials, alongside some foundational results. 2. **Extensions de corps**: Focused on algebraic closure and the primitive element lemma, this section lays the groundwork for understanding field theory. 3. **Polynôme minimal**: Here, students engage with concepts of conjugate elements, which are crucial for grasping the broader implications of field extensions. 4. **Corps fini**: In this section, Frobenius automorphisms and extensions of finite fields are explored, enriching the discussion on field theory. 5. **Théorie des groupes I**: Key results about group theory are presented, including Lagrange's theorem and the order of an element—essential tools for understanding Galois groups. 6. **Correspondance de Galois**: The Artin lemma and Galois correspondence are examined in depth, establishing a clear link between field extensions and their respective Galois groups. 7. **Théorie des groupes II**: Diving deeper, this module covers solvable groups and provides a rigorous proof of the non-solvability of the symmetric group \(S_n\) for \(n \geq 5\). 8. **Cyclotomie I**: Students will study general cyclotomic extensions and Kummer theory, bridging the gaps between theory and application. 9. **Théorèmes de résolubilité de Galois**: The course concludes with critical criteria for solvability and a theorem focused on specific degrees, enabling learners to wrap up their understanding of the key results in Galois theory. 10. **Réduction mod p**: This segment investigates Galois groups of polynomials with integer coefficients under modulo \(p\) reduction, further broadening the context of polynomial solvability. 11. **Compléments**: Offering additional insights, this section touches on cyclotomic properties over \(\mathbb{Q}\) and other significant applications that stem from Galois theory. ### Review and Recommendation This course is highly recommended for mathematics enthusiasts, particularly those with a keen interest in abstract algebra. The structured approach allows learners to gradually navigate through complex concepts, making it accessible for those with a basic understanding of algebra. The course's depth ensures that participants not only learn the theory behind Galois but also appreciate its practical applications in modern mathematical contexts. The teaching style is engaging, with clear explanations and numerous examples that help cement understanding. Furthermore, the use of real-world applications enhances the relevance of Galois theory concepts, inspiring students to appreciate its significance in the broader scope of mathematics. In summary, "Introduction à la théorie de Galois" is an essential course for anyone serious about expanding their knowledge in algebra and exploring the rich tapestry of polynomial roots and their symmetries. Whether you are a student, educator, or simply a math enthusiast, this course offers invaluable insights into one of the most beautiful areas of mathematics. Enroll today, and embark on a transformative journey through the enchanting world of Galois theory!

Syllabus

Introduction

description du problème et quelques résultats sur les polynômes d'une variable comme échauffement

algébricité, corps algébriquement clos, lemme de l'élément primitif

éléments conjugués

Frobenius, automorphismes, extensions de corps finis

résultats de base, ordre d’un élément, théorème de Lagrange

lemme d'Artin, groupes de Galois, correspondance de Galois

groupes résolubles, non résolubilité du groupe symétrique Sn pour n plus grand ou égal à 5

extension cyclotomique générale, théorie de Kummer

critère de résolubilité, théorème de Galois en degré p

groupes de Galois de polynômes à coefficients entiers par réduction modulo p

Cyclotomie sur Q (grâce à la réduction modulo p) et autres applications.