課程信息
4.3
91 個評分
26 個審閱
100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
高級

高級

完成時間(小時)

完成時間大約為45 小時

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

英語(English)

字幕:英語(English)...
100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
高級

高級

完成時間(小時)

完成時間大約為45 小時

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

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(小時)
完成時間為 23 分鐘

Introduction

This is just a two-minutes advertisement and a short reference list....
Reading
1 個視頻(共 3 分鐘), 2 個閱讀材料
Video1 個視頻
Reading2 個閱讀材料
Introduction/Manual10分鐘
References10分鐘
完成時間(小時)
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....
Reading
6 個視頻(共 84 分鐘), 1 個測驗
Video6 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
Quiz1 個練習
Quiz 112分鐘
2
完成時間(小時)
完成時間為 1 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....
Reading
5 個視頻(共 67 分鐘), 1 個測驗
Video5 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
Quiz1 個練習
QUIZ 212分鐘
3
完成時間(小時)
完成時間為 2 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. ...
Reading
6 個視頻(共 82 分鐘), 1 個閱讀材料, 1 個測驗
Video6 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
Reading1 個閱讀材料
Ungraded assignment 110分鐘
Quiz1 個練習
QUIZ 38分鐘
4
完成時間(小時)
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")....
Reading
6 個視頻(共 91 分鐘), 1 個測驗
Video6 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
Quiz1 個練習
QUIZ 410分鐘
4.3

熱門審閱

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

Avatar

Ekaterina Amerik

Professor
Department of Mathematics

關於 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...

常見問題

  • 注册以便获得证书后,您将有权访问所有视频、测验和编程作业(如果适用)。只有在您的班次开课之后,才可以提交和审阅同学互评作业。如果您选择在不购买的情况下浏览课程,可能无法访问某些作业。

  • 您购买证书后,将有权访问所有课程材料,包括评分作业。完成课程后,您的电子课程证书将添加到您的成就页中,您可以通过该页打印您的课程证书或将其添加到您的领英档案中。如果您只想阅读和查看课程内容,可以免费旁听课程。

還有其他問題嗎?請訪問 學生幫助中心