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Ekaterina Amerik

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Introduction to Galois Theory

俄罗斯国家研究型高等经济大学

课程信息

7,917 次近期查看

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...)


Do you have technical problems? Write to us: coursera@hse.ru

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高级

完成时间大约为27 小时

英语（English）

字幕：法语（French）, （欧洲人讲的）葡萄牙语, 俄语（Russian）, 英语（English）, 西班牙语（Spanish）

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根据您的日程表重置截止日期。

可分享的证书

完成后获得证书

100% 在线

立即开始，按照自己的计划学习。

高级

完成时间大约为27 小时

英语（English）

字幕：法语（French）, （欧洲人讲的）葡萄牙语, 俄语（Russian）, 英语（English）, 西班牙语（Spanish）

授课教师

授课教师评分

4.5/5 （9 个评分）

Ekaterina Amerik

Professor

Department of Mathematics

24,427 个学生

1 门课程

提供方

俄罗斯国家研究型高等经济大学

HSE University 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 communicamathematics, engineering, and more.

授课大纲 - 您将从这门课程中学到什么

周

周 1

完成时间为 1 小时

Introduction

This is just a two-minutes advertisement and a short reference list.

完成时间为 1 小时

2 个视频（总计 4 分钟）, 4 个阅读材料

完成时间为 2 小时

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.

完成时间为 2 小时

6 个视频（总计 84 分钟）

周

周 2

完成时间为 2 小时

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.

完成时间为 2 小时

5 个视频（总计 67 分钟）

周

周 3

完成时间为 4 小时

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.

完成时间为 4 小时

6 个视频（总计 82 分钟）, 1 个阅读材料, 1 个测验

周

周 4

完成时间为 2 小时

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").

完成时间为 2 小时

6 个视频（总计 91 分钟）

周

周 5

完成时间为 4 小时

Week 5

We apply the discussion from the last lecture to the case of field extensions. We show that the separable extensions remain reduced after a base change: the inseparability is responsible for eventual nilpotents. As our next subject, we introduce normal and Galois extensions and prove Artin's theorem on invariants. This week, the first graded assignment is given.

完成时间为 4 小时

6 个视频（总计 81 分钟）

周

周 6

完成时间为 2 小时

Week 6

We state and prove the main theorem of these lectures: the Galois correspondence. Then we start doing examples (low degree, discriminant, finite fields, roots of unity).

完成时间为 2 小时

6 个视频（总计 86 分钟）

6 个视频

6.1 Some further remarks on normal extension. Fixed field.13分钟

6.2 The Galois correspondence14分钟

6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.14分钟

6.4 Discriminant. Degree 3 (cont'd). Finite fields.15分钟

6.5 An infinite degree example. Roots of unity: cyclotomic polynomials14分钟

6.6 Irreducibility of cyclotomic polynomial.The Galois group.14分钟

1 个练习

QUIZ 640分钟

周

周 7

完成时间为 4 小时

Week 7

We continue to study the examples: cyclotomic extensions (roots of unity), cyclic extensions (Kummer and Artin-Schreier extensions). We introduce the notion of the composite extension and make remarks on its Galois group (when it is Galois), in the case when the composed extensions are in some sense independent and one or both of them is Galois. The notion of independence is also given a precise sense ("linearly disjoint extensions"). This week, an ungraded assignment is given.

完成时间为 4 小时

7 个视频（总计 87 分钟）, 1 个阅读材料

7 个视频

7.1 Cyclotomic extensions (cont'd). Examples over Q.12分钟

7.2. Kummer extensions.14分钟

7.3. Artin-Schreier extensions.11分钟

7.4. Composite extensions. Properties.13分钟

7.5. Linearly disjoint extensions. Examples.15分钟

7.6. Linearly disjoint extensions in the Galois case.12分钟

7.7 On the Galois group of the composite.7分钟

1 个阅读材料

Ungraded assignment 22 小时 5 分

周

周 8

完成时间为 2 小时

Week 8

We finally arrive to the source of Galois theory, the question which motivated Galois himself: which equation are solvable by radicals and which are not? We explain Galois' result: an equation is solvable by radicals if and only if its Galois group is solvable in the sense of group theory. In particular we see that the "general" equation of degree at least 5 is not solvable by radicals. We briefly discuss the relations to representation theory and to topological coverings.

完成时间为 2 小时

6 个视频（总计 81 分钟）

6 个视频

8.1. Extensions solvable by radicals. Solvable groups. Example.15分钟

8.2. Properties of solvable groups. Symmetric group.13分钟

8.3.Galois theorem on solvability by radicals.11分钟

8.4.Examples of equations not solvable by radicals."General equation".13分钟

8.5. Galois action as a representation. Normal base theorem.14分钟

8.6. Normal base theorem (cont'd). Relation with coverings.12分钟

1 个练习

QUIZ 840分钟

周

周 9

完成时间为 4 小时

Week 9.

We build a tool for finding elements in Galois groups, learning to use the reduction modulo p. For this, we have to talk a little bit about integral ring extensions and also about norms and traces.This week, the final graded assignment is given.

完成时间为 4 小时

6 个视频（总计 84 分钟）

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来自INTRODUCTION TO GALOIS THEORY的热门评论

由 MP 提供Jul 30, 2020

A difficult course for me, personally, but that makes it all the more worth it! Taking this course has helped me learn more I thought I would. Definitely recommended.

由 F 提供Sep 11, 2021

Thank You Prof Ekaterina Amerik for this course. I understand with your explaination. Maybe i must study again and again for understand this material.

由 CL 提供Jun 15, 2016

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

由 TW 提供Mar 11, 2018

The teacher is good at explaining things. It is best you take an algebra course for prerequisite.

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Introduction to Galois Theory

Introduction to Galois Theory

课程信息