このコースについて
6,116

100%オンライン

自分のスケジュールですぐに学習を始めてください。

柔軟性のある期限

スケジュールに従って期限をリセットします。

上級レベル

約45時間で修了

推奨:9 weeks of study, 4-8 hours/week...

英語

字幕:英語

100%オンライン

自分のスケジュールですぐに学習を始めてください。

柔軟性のある期限

スケジュールに従って期限をリセットします。

上級レベル

約45時間で修了

推奨:9 weeks of study, 4-8 hours/week...

英語

字幕:英語

シラバス - 本コースの学習内容

1
23分で修了

Introduction

This is just a two-minutes advertisement and a short reference list....
1件のビデオ (合計3分), 2 readings
1件のビデオ
2件の学習用教材
Introduction/Manual10 分
References10 分
2時間で修了

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....
6件のビデオ (合計84分), 1 quiz
6件のビデオ
1.2 Algebraic elements. Minimal polynomial.12 分
1.3 Algebraic elements. Algebraic extensions.14 分
1.4 Finite extensions. Algebraicity and finiteness.14 分
1.5 Algebraicity in towers. An example.14 分
1.6. A digression: Gauss lemma, Eisenstein criterion.13 分
1の練習問題
Quiz 140 分
2
2時間で修了

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....
5件のビデオ (合計67分), 1 quiz
5件のビデオ
2.2 Splitting field.11 分
2.3 An example. Algebraic closure.14 分
2.4 Algebraic closure (continued).15 分
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.11 分
1の練習問題
QUIZ 240 分
3
4時間で修了

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. ...
6件のビデオ (合計82分), 1 reading, 1 quiz
6件のビデオ
3.2 Properties of finite fields.14 分
3.3 Multiplicative group and automorphism group of a finite field.15 分
3.4 Separable elements.15 分
3.5. Separable degree, separable extensions.15 分
3.6 Perfect fields.9 分
1件の学習用教材
Ungraded assignment 1
1の練習問題
QUIZ 340 分
4
2時間で修了

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")....
6件のビデオ (合計91分), 1 quiz
6件のビデオ
4.2 Tensor product of modules14 分
4.3 Base change14 分
4.4 Examples. Tensor product of algebras.15 分
4.5 Relatively prime ideals. Chinese remainder theorem.14 分
4.6 Structure of finite algebras over a field. Examples.16 分
1の練習問題
QUIZ 440 分
5
4時間で修了

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....
6件のビデオ (合計81分), 2 quizzes
6件のビデオ
5.2 Separability and base change14 分
5.3 Separability and base change (cont'd). Primitive element theorem.14 分
5.4 Examples. Normal extensions.13 分
5.5 Galois extensions.11 分
5.6 Artin's theorem.13 分
1の練習問題
QUIZ 540 分
6
2時間で修了

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)....
6件のビデオ (合計86分), 1 quiz
6件のビデオ
6.2 The Galois correspondence14 分
6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.14 分
6.4 Discriminant. Degree 3 (cont'd). Finite fields.15 分
6.5 An infinite degree example. Roots of unity: cyclotomic polynomials14 分
6.6 Irreducibility of cyclotomic polynomial.The Galois group.14 分
1の練習問題
QUIZ 640 分
7
4時間で修了

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....
7件のビデオ (合計87分), 1 reading
7件のビデオ
7.2. Kummer extensions.14 分
7.3. Artin-Schreier extensions.11 分
7.4. Composite extensions. Properties.13 分
7.5. Linearly disjoint extensions. Examples.15 分
7.6. Linearly disjoint extensions in the Galois case.12 分
7.7 On the Galois group of the composite.7 分
1件の学習用教材
Ungraded assignment 25 分
8
2時間で修了

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....
6件のビデオ (合計81分), 1 quiz
6件のビデオ
8.2. Properties of solvable groups. Symmetric group.13 分
8.3.Galois theorem on solvability by radicals.11 分
8.4.Examples of equations not solvable by radicals."General equation".13 分
8.5. Galois action as a representation. Normal base theorem.14 分
8.6. Normal base theorem (cont'd). Relation with coverings.12 分
1の練習問題
QUIZ 840 分
9
4時間で修了

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....
6件のビデオ (合計84分), 2 quizzes
6件のビデオ
9.2. Integral extensions, integral closure, ring of integers of a number field.15 分
9.3. Norm and trace.14 分
9.4. Norm and trace (cont'd). Ring of integers is a free module.13 分
9.5. Reduction modulo a prime.13 分
9.6. Reduction modulo a prime and finding elements in Galois groups.14 分
1の練習問題
QUIZ 940 分
4.3
27件のレビューChevron Right

人気のレビュー

by 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.

講師

Avatar

Ekaterina Amerik

Professor
Department of Mathematics

ロシア国立研究大学経済高等学院(National Research University Higher School of Economics)について

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 communicamathematics, engineering, and more. Learn more on www.hse.ru...

よくある質問

  • 修了証に登録すると、すべてのビデオ、テスト、およびプログラミング課題(該当する場合)にアクセスできます。ピアレビュー課題は、セッションが開始してからのみ、提出およびレビューできます。購入せずにコースを検討することを選択する場合、特定の課題にアクセスすることはできません。

  • 修了証を購入する際、コースのすべての教材(採点課題を含む)にアクセスできます。コースを完了すると、電子修了証が成果のページに追加されます。そこから修了証を印刷したり、LinkedInのプロフィールに追加したりできます。コースの内容の閲覧のみを希望する場合は、無料でコースを聴講できます。

さらに質問がある場合は、受講者向けヘルプセンターにアクセスしてください。