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高級

完成時間大約為45 小時

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

英語(English)

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立即開始,按照自己的計劃學習。

可靈活調整截止日期

根據您的日程表重置截止日期。

高級

完成時間大約為45 小時

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

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 23 分鐘

Introduction

This is just a two-minutes advertisement and a short reference list....
1 個視頻 (總計 3 分鐘), 2 個閱讀材料
1 個視頻
2 個閱讀材料
Introduction/Manual10分鐘
References10分鐘
完成時間為 2 小時

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 個視頻 (總計 84 分鐘), 1 個測驗
6 個視頻
1.2 Algebraic elements. Minimal polynomial.12分鐘
1.3 Algebraic elements. Algebraic extensions.14分鐘
1.4 Finite extensions. Algebraicity and finiteness.14分鐘
1.5 Algebraicity in towers. An example.14分鐘
1.6. A digression: Gauss lemma, Eisenstein criterion.13分鐘
1 個練習
Quiz 140分鐘
2
完成時間為 2 小時

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 個視頻 (總計 67 分鐘), 1 個測驗
5 個視頻
2.2 Splitting field.11分鐘
2.3 An example. Algebraic closure.14分鐘
2.4 Algebraic closure (continued).15分鐘
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.11分鐘
1 個練習
QUIZ 240分鐘
3
完成時間為 4 小時

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 個視頻 (總計 82 分鐘), 1 個閱讀材料, 1 個測驗
6 個視頻
3.2 Properties of finite fields.14分鐘
3.3 Multiplicative group and automorphism group of a finite field.15分鐘
3.4 Separable elements.15分鐘
3.5. Separable degree, separable extensions.15分鐘
3.6 Perfect fields.9分鐘
1 個閱讀材料
Ungraded assignment 1
1 個練習
QUIZ 340分鐘
4
完成時間為 2 小時

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 個視頻 (總計 91 分鐘), 1 個測驗
6 個視頻
4.2 Tensor product of modules14分鐘
4.3 Base change14分鐘
4.4 Examples. Tensor product of algebras.15分鐘
4.5 Relatively prime ideals. Chinese remainder theorem.14分鐘
4.6 Structure of finite algebras over a field. Examples.16分鐘
1 個練習
QUIZ 440分鐘
5
完成時間為 4 小時

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 個視頻 (總計 81 分鐘), 2 個測驗
6 個視頻
5.2 Separability and base change14分鐘
5.3 Separability and base change (cont'd). Primitive element theorem.14分鐘
5.4 Examples. Normal extensions.13分鐘
5.5 Galois extensions.11分鐘
5.6 Artin's theorem.13分鐘
1 個練習
QUIZ 540分鐘
6
完成時間為 2 小時

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 個視頻 (總計 86 分鐘), 1 個測驗
6 個視頻
6.2 The Galois correspondence14分鐘
6.3 Galois correspondence (cont'd). First examples (polynomials of degree 2 and 3.14分鐘
6.4 Discriminant. Degree 3 (cont'd). Finite fields.15分鐘
6.5 An infinite degree example. Roots of unity: cyclotomic polynomials14分鐘
6.6 Irreducibility of cyclotomic polynomial.The Galois group.14分鐘
1 個練習
QUIZ 640分鐘
7
完成時間為 4 小時

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 個視頻 (總計 87 分鐘), 1 個閱讀材料
7 個視頻
7.2. Kummer extensions.14分鐘
7.3. Artin-Schreier extensions.11分鐘
7.4. Composite extensions. Properties.13分鐘
7.5. Linearly disjoint extensions. Examples.15分鐘
7.6. Linearly disjoint extensions in the Galois case.12分鐘
7.7 On the Galois group of the composite.7分鐘
1 個閱讀材料
Ungraded assignment 25分鐘
8
完成時間為 2 小時

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 個視頻 (總計 81 分鐘), 1 個測驗
6 個視頻
8.2. Properties of solvable groups. Symmetric group.13分鐘
8.3.Galois theorem on solvability by radicals.11分鐘
8.4.Examples of equations not solvable by radicals."General equation".13分鐘
8.5. Galois action as a representation. Normal base theorem.14分鐘
8.6. Normal base theorem (cont'd). Relation with coverings.12分鐘
1 個練習
QUIZ 840分鐘
9
完成時間為 4 小時

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 個視頻 (總計 84 分鐘), 2 個測驗
6 個視頻
9.2. Integral extensions, integral closure, ring of integers of a number field.15分鐘
9.3. Norm and trace.14分鐘
9.4. Norm and trace (cont'd). Ring of integers is a free module.13分鐘
9.5. Reduction modulo a prime.13分鐘
9.6. Reduction modulo a prime and finding elements in Galois groups.14分鐘
1 個練習
QUIZ 940分鐘
4.3
27 個審閱Chevron Right

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創建者 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

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Ekaterina Amerik

Professor
Department of Mathematics

關於 国立高等经济大学

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...

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