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課程名稱 |
代數一
ALGEBRA(Ⅰ) |
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開課學期 |
99-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
林惠雯 |
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課號 |
MATH7105 |
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課程識別碼 |
221 U3830 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) |
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上課地點 |
新304新502 |
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備註 |
研究所數學組基礎課。先修知識:代數導論
總人數上限：60人
外系人數限制：5人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/991abstract_algebra |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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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. Group theory:
Sylow theorems, classification of finite groups of simple type, solvable groups, free groups, simple groups.
2. Ring theory:
Rings of fractions, ED, PID, UFD, Resultant, Polynomial rings, Grobner basis.
3. Field theory:
Field extensions, the fundamental theorem of Galois theory, finite fields,solution by radicals, Abelian extension, Kummer extension. |
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課程目標 |
Course Goal：
For students who are interested in various related fields, we hope to equip students with a solid foundation in algebra.
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課程要求 |
建議先修:Undergraduate algebra.
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預期每週課後學習時數 |
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Office Hours |
另約時間 |
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指定閱讀 |
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參考書目 |
Dummit-Foote:Abstract Algebra |
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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 |
30% |
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2. |
Midterm |
35% |
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3. |
Final |
35% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/13,9/16 |
Quotient groups/3 isomorphism theorems |
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第2週 |
9/20,9/23 |
Group actions/Sylow theorems; 作業: Section 4.2: 8, 10, 11 /Section 4.3: 5, 12, 23, 24, 27, 34 (加分題:30) |
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第3週 |
9/27,9/30 |
Simplicity of A_n/Direct product + 助教習題課; 作業: Section 4.5: 13,16,27,32,34 (加分題:44) /Section 5.2: 3,9,15 |
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第4週 |
10/04,10/07 |
Classification of groups of order 30,12,p^3/solvable groups(I) + 助教習題課; 作業: Section 5.4: 10, 15 + Section 5.5: 6,8,10 (choose two of them)/ Section 6.1: 3, 4 |
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第5週 |
10/11,10/14 |
Solvable groups(II)/Free groups(I) + 助教習題課; 作業: Section 6.1: 7,9,10,20/Section 6.3: 1,8,11 |
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第6週 |
10/18,10/21 |
Free group(II) + simple groups/ Chinese Remainder theorem + 助教習題課; 作業: Section 6.2: 1, 3, 7, 22/Section 7.6: 3, 6, 7 |
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第7週 |
10/25,10/28 |
Rings of fractions /ED,PID,UFD; 作業: Section 7.5: 3, 4, 5/Section 8.2: 4, 6 |
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第8週 |
11/01,11/04 |
Rings of quadratic algebraic integers/resultant + Gauss lemma; 作業: Section 8.1: 7, 10 & Section 8.3: 6/Section 14.6: 35 & Section 9.3: 2, 4 |
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第9週 |
11/08,11/11 |
Irreducibility + 助教習題課/期中考 作業: Section 9.4: 3, 12 |
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第10週 |
11/15,11/18 |
校慶停課/Grobner basis(I); 作業: Section 9.6: 10, 11, 12, 14 |
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第11週 |
11/22,11/25 |
Grobner basis(II)+finite extension/simple extension); 作業: Section 9.6: 19, 23, 32/Section 13.1: 1, 5, 6, 7 |
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第12週 |
11/29,12/02 |
Algebraic extension/splitting fields); 作業: Section 13.2: 7, 11, 13, 14/Section 13.4: 3,6 |
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第13週 |
12/06,12/09 |
Algebraic closure + separable extension(I)/separable extension(II); 作業: Section 13.6: 2, 3, 6, 15/Section 13.5: 2, 5, 7, 11 |
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第14週 |
12/13,12/16 |
Galois extension/助教習題課(參加ICCM); 作業: Section 14.1: 1, 4, 10; Section 14.2: 5, 14, 16 |
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第15週 |
12/20,12/23 |
助教習題課(參加ICCM)/Fundamental theorem of Galois theory; 作業: Section 14.3: 3, 4, 10 |
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第16週 |
12/27,12/30 |
Finite fields, Abelian extension/Abelian extension,Kummer extension); 作業: Section 14.5:1, 11/Section 14.4: 2, 6 |
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第17週 |
1/03,1/06 |
Kummer extension,solution by radicals/solution by radicals,助教習題課 |
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第18週 |
1/10,1/13 |
/期末考 |
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