Abstract algebra – Topics for talks

Instructor: M. Grützmann (古梅西)
Textbook: P.A. Grillet, Abstract Algebra, GTM, Springer (2007), ISBN 978-0-387-71568-1. (Here is an online version.1)
Course home page: www.grutzmann.de/algebra.

I. Groups (5+ weeks)

1.4. Free groups (自由群), free products (自由积), and presentations (群的展示)

1.5 Direct products (真积) - 王陆华

1.6 The Krull–Schmidt theorem (克鲁尔–施密特定理) – 张阅化

1.8 Structure of symmetric groups (置换群) – 杨若松

1.10 Small groups (分类的小群) – 曹宁

1.11 The general linear group (一般线性群) – 朱翠影

II. Rings and algebras (换理论与代数, 3 weeks)

2.2 Homomorphisms (同态), subrings (子环), and ideals (理想) – 荆慧双

2.3 Domains (整环) and fields (域) – 代菡

2.4 Principal ideal domains (主理想环) – 刘意

2.5 Unique factorization domains (唯一分解整环) – 郝志香

III Field extensions (域扩张) and Galois theory (伽罗瓦理论, 5 weeks)

3.3 Separable extensions (可分扩张) – 王会菊

3.4 Resultants (结式) and discriminants (判别式) – 周Rui

3.6 Galois extensions (伽罗瓦扩张) and the correspondence principle (对应原理) – Cheng明宏

3.8 Cyclotomic (分圆), cyclic extensions (循环扩张) and Solvability by radicals (可解用根式) – 武乐云

3.10 Geometric constructions (尺规作图) – 郭欠霞

3.12 Outlook: Algebraic geometry (代数几何) – 何挺

Further literature

  1. M. Artin: Algebra, 2nd edition, Addison Wesley (2010), ISBN 978-013-241-377-0.
  2. N. Jacobson: Lectures in abstract algebra: 1 basic concepts, 3 Fields and Galois theory, Springer, (2000), ISBN 750-620-060-0, 750-620-062-7.
  3. 里克正 (Li Kezheng): 抽象代数基础 (Basic Algebra), Springer (2007)/ Higher Education Press, ISBN 978-730-214-407-6.
  4. J.J. Rothman: A first course in abstract algebra, 3rd edition, Prentice Hall (2006), ISBN 978-711-121-262-1.
  5. 杨子胥 (Yang Zixu): 近世代数 (International Algebra), 3rd edition, Higher Education Press (2011), ISBN 978-704-030-072-7.

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