À propos de ce cours
4.3
90 notes
26 avis
A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions. We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail. After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.). We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups. PREREQUISITES A first course in general algebra — groups, rings, fields, modules, ideals. Some knowledge of commutative algebra (prime and maximal ideals — first few pages of any book in commutative algebra) is welcome. For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally, the statement of Sylow's theorems. ASSESSMENTS A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count approximately 30%, first (shorter) exam 30%, final exam 40%. There will be two non-graded exercise lists (in replacement of the non-existent exercise classes...)...
Globe

Cours en ligne à 100 %

Commencez dès maintenant et apprenez aux horaires qui vous conviennent.
Calendar

Dates limites flexibles

Réinitialisez les dates limites selon votre disponibilité.
Advanced Level

Niveau avancé

Clock

Approx. 43 hours to complete

Recommandé : 9 weeks of study, 4-8 hours/week...
Comment Dots

English

Sous-titres : English...
Globe

Cours en ligne à 100 %

Commencez dès maintenant et apprenez aux horaires qui vous conviennent.
Calendar

Dates limites flexibles

Réinitialisez les dates limites selon votre disponibilité.
Advanced Level

Niveau avancé

Clock

Approx. 43 hours to complete

Recommandé : 9 weeks of study, 4-8 hours/week...
Comment Dots

English

Sous-titres : English...

Programme du cours : ce que vous apprendrez dans ce cours

Week
1
Clock
23 minutes pour terminer

Introduction

This is just a two-minutes advertisement and a short reference list....
Reading
1 vidéo (Total 3 min), 2 lectures
Reading2 lectures
Introduction/Manual10 min
References10 min
Clock
2 heures pour terminer

Week 1

We introduce the basic notions such as a field extension, algebraic element, minimal polynomial, finite extension, and study their very basic properties such as the multiplicativity of degree in towers....
Reading
6 vidéos (Total 84 min), 1 quiz
Video6 vidéos
1.2 Algebraic elements. Minimal polynomial.12 min
1.3 Algebraic elements. Algebraic extensions.14 min
1.4 Finite extensions. Algebraicity and finiteness.14 min
1.5 Algebraicity in towers. An example.14 min
1.6. A digression: Gauss lemma, Eisenstein criterion.13 min
Quiz1 exercice pour s'entraîner
Quiz 112 min
Week
2
Clock
1 heure pour terminer

Week 2

We introduce the notion of a stem field and a splitting field (of a polynomial). Using Zorn's lemma, we construct the algebraic closure of a field and deduce its unicity (up to an isomorphism) from the theorem on extension of homomorphisms....
Reading
5 vidéos (Total 67 min), 1 quiz
Video5 vidéos
2.2 Splitting field.11 min
2.3 An example. Algebraic closure.14 min
2.4 Algebraic closure (continued).15 min
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.11 min
Quiz1 exercice pour s'entraîner
QUIZ 212 min
Week
3
Clock
2 heures pour terminer

Week 3

We recall the construction and basic properties of finite fields. We prove that the multiplicative group of a finite field is cyclic, and that the automorphism group of a finite field is cyclic generated by the Frobenius map. We introduce the notions of separable (resp. purely inseparable) elements, extensions, degree. We briefly discuss perfect fields. This week, the first ungraded assignment (in order to practice the subject a little bit) is given. ...
Reading
6 vidéos (Total 82 min), 1 lecture, 1 quiz
Video6 vidéos
3.2 Properties of finite fields.14 min
3.3 Multiplicative group and automorphism group of a finite field.15 min
3.4 Separable elements.15 min
3.5. Separable degree, separable extensions.15 min
3.6 Perfect fields.9 min
Reading1 lecture
Ungraded assignment 110 min
Quiz1 exercice pour s'entraîner
QUIZ 38 min
Week
4
Clock
2 heures pour terminer

Week 4

This is a digression on commutative algebra. We introduce and study the notion of tensor product of modules over a ring. We prove a structure theorem for finite algebras over a field (a version of the well-known "Chinese remainder theorem")....
Reading
6 vidéos (Total 91 min), 1 quiz
Video6 vidéos
4.2 Tensor product of modules14 min
4.3 Base change14 min
4.4 Examples. Tensor product of algebras.15 min
4.5 Relatively prime ideals. Chinese remainder theorem.14 min
4.6 Structure of finite algebras over a field. Examples.16 min
Quiz1 exercice pour s'entraîner
QUIZ 410 min
4.3

Meilleurs avis

par CLJun 16th 2016

Outstanding course so far - a great refresher for me on Galois theory. It's nice to see more advanced mathematics classes on Coursera.

Enseignant

Ekaterina Amerik

Professor
Department of Mathematics

À propos de National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

Foire Aux Questions

  • Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

  • When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

D'autres questions ? Visitez le Centre d'Aide pour les Etudiants.