课程信息
4.3
91 个评分
26 个审阅
100% 在线

100% 在线

立即开始,按照自己的计划学习。
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可灵活调整截止日期

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高级

高级

完成时间(小时)

完成时间大约为45 小时

建议:9 weeks of study, 4-8 hours/week...
可选语言

英语(English)

字幕:英语(English)
100% 在线

100% 在线

立即开始,按照自己的计划学习。
可灵活调整截止日期

可灵活调整截止日期

根据您的日程表重置截止日期。
高级

高级

完成时间(小时)

完成时间大约为45 小时

建议:9 weeks of study, 4-8 hours/week...
可选语言

英语(English)

字幕:英语(English)

教学大纲 - 您将从这门课程中学到什么

1
完成时间(小时)
完成时间为 23 分钟

Introduction

This is just a two-minutes advertisement and a short reference list....
Reading
1 个视频 (总计 3 分钟), 2 个阅读材料
Video1 个视频
Reading2 个阅读材料
Introduction/Manual10分钟
References10分钟
完成时间(小时)
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....
Reading
6 个视频 (总计 84 分钟), 1 个测验
Video6 个视频
1.2 Algebraic elements. Minimal polynomial.12分钟
1.3 Algebraic elements. Algebraic extensions.14分钟
1.4 Finite extensions. Algebraicity and finiteness.14分钟
1.5 Algebraicity in towers. An example.14分钟
1.6. A digression: Gauss lemma, Eisenstein criterion.13分钟
Quiz1 个练习
Quiz 112分钟
2
完成时间(小时)
完成时间为 1 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....
Reading
5 个视频 (总计 67 分钟), 1 个测验
Video5 个视频
2.2 Splitting field.11分钟
2.3 An example. Algebraic closure.14分钟
2.4 Algebraic closure (continued).15分钟
2.5 Extension of homomorphisms. Uniqueness of algebraic closure.11分钟
Quiz1 个练习
QUIZ 212分钟
3
完成时间(小时)
完成时间为 2 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. ...
Reading
6 个视频 (总计 82 分钟), 1 个阅读材料, 1 个测验
Video6 个视频
3.2 Properties of finite fields.14分钟
3.3 Multiplicative group and automorphism group of a finite field.15分钟
3.4 Separable elements.15分钟
3.5. Separable degree, separable extensions.15分钟
3.6 Perfect fields.9分钟
Reading1 个阅读材料
Ungraded assignment 110分钟
Quiz1 个练习
QUIZ 38分钟
4
完成时间(小时)
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")....
Reading
6 个视频 (总计 91 分钟), 1 个测验
Video6 个视频
4.2 Tensor product of modules14分钟
4.3 Base change14分钟
4.4 Examples. Tensor product of algebras.15分钟
4.5 Relatively prime ideals. Chinese remainder theorem.14分钟
4.6 Structure of finite algebras over a field. Examples.16分钟
Quiz1 个练习
QUIZ 410分钟
4.3
26 个审阅Chevron Right

热门审阅

创建者 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.

创建者 RZMay 15th 2016

Very interesting course for poeple with knowledge in algebra.

讲师

Avatar

Ekaterina Amerik

Professor
Department of Mathematics

关于 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...

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