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課程名稱 |
近世代數二
Modern Algebra (Ⅱ) |
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開課學期 |
109-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
莊武諺 |
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課號 |
MATH5002 |
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課程識別碼 |
221 U6590 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四8,9(15:30~17:20) |
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上課地點 |
天數305天數305 |
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備註 |
總人數上限：30人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1092algebra2 |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
1. Commutative algebras.
2. Homological algebras.
3. Field theory. |
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課程目標 |
Provide students with fundamentals in algebra. |
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課程要求 |
Prerequisite: undergraduate algebra, graduate algebra I |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Atiyah and MacDonald, Introduction to Commutative Algebra.
Gelfand and Manin, Methods of Homological Algebra.
Milne, Fields and Galois Theory. |
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參考書目 |
待補 |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
40% |
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2. |
Midterm |
30% |
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3. |
Final |
30% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/22,2/25 |
Chain conditions. |
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第2週 |
3/01,3/04 |
Noetherian rings. |
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第3週 |
3/08,3/11 |
Artin rings. |
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第4週 |
3/15,3/18 |
DVR, Dedekind domains. |
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第5週 |
3/22,3/25 |
Completions. |
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第6週 |
3/29,4/01 |
Dimension theory. 4/1: no class. |
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第7週 |
4/05,4/08 |
Dimension theory. 4/5: no class. |
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第8週 |
4/12,4/15 |
4/12: Basics of field theory. 4/15: midterm. |
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第9週 |
4/19,4/22 |
Splitting fields, multiple roots. |
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第10週 |
4/26,4/29 |
Galois theory. |
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第11週 |
5/03,5/06 |
Galois groups of cubic and quartic polynomials, applications of Galois theory. |
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第12週 |
5/10,5/13 |
5/10: Insovability.
5/13: triangulated spaces, homology and cohomology. |
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第13週 |
5/17,5/20 |
Note1: sheaves, germs and fibers, sheafification, Cech complex, homotopy of complex morphisms. |
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第14週 |
5/24,5/27 |
Note2: category theory, category of functors, equivalence of categories.
Note3: equivalence of categories, representable functors, Yoneda lemma, adjoint functors. |
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第15週 |
5/31,6/03 |
Note4: adjoint morphisms, additive and abelian categories.
Note5: functors in abelian categories, projective modules, injective modules, flat modules. |
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第16週 |
6/07,6/10 |
Note6: direct image, inverse image.Note7: complexes as objects, uniqueness of resolutions, very short introduction to derived categories. |
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