課程名稱 |
代數導論優一 Honors Algebra (Ⅰ) |
開課學期 |
100-1 |
授課對象 |
理學院 數學系 |
授課教師 |
林惠雯 |
課號 |
MATH2109 |
課程識別碼 |
201 49450 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) |
上課地點 |
天數101天數101 |
備註 |
1.需修過微積分及線性代數,且分數達B以上或其中一科達B+以上。2.代數導論優可抵必修代數導論。 總人數上限:50人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1001Honors_algebra |
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 |
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:
The theory of groups is one of the oldest and richest branches of algebra.
(1) Explanation of fundamental concepts:
Permutation groups, Cyclic groups, Group homomorphisms, Quotient groups, Group actions;
(2) Finite groups being the basis of Galois' discoveries in the theory of equations:
Sylow theorems, Classification of finite groups, Simple groups, Solvable groups.
2. Ring theory:
The theory of rings grew out of the study of two particular classes of rings, polynomial rings and the "integers" of an algebraic number field.
(1) Basic concepts concerning rings:
Ideals, Rings of fractions, Euclidean domains, Principal ideal domains, Unique factorization domains;
(2) Standard examples:
Matrix rings, Polynomial rings, Rings of quadratic algebraic integers;
(3) Computational techniques enhancing the development:
Resultant,Grobner basis.
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課程目標 |
Fundamental to all areas of mathematics, algebra provides the cornerstone for the student's development. In this course, in addition to the basic concepts, advanced material will be introduced. We would like to give students an insight into more advanced algebraic topics. |
課程要求 |
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預期每週課後學習時數 |
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Office Hours |
備註: 星期四 12:20 ~ 1:20 |
指定閱讀 |
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參考書目 |
Textbook :
N. Jacobson, Basic Algebra I , 2nd edition
References :
M. Artin, Algebra, 2nd edition
Dummit-Foote, Abstract Algebra
Serge Lang, Undergraduate Algebra, 3rd edition
B.L. van der Waerden, Algebra I,7th edition
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評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Homework and quiz |
35% |
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2. |
Middle examination |
35% |
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3. |
Final examination |
30% |
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週次 |
日期 |
單元主題 |
第1週 |
9/12, 9/15 |
中秋節放假/Definition of groups,examples |
第2週 |
9/19, 9/22 |
Cayley's theorem/Generators of groups |
第3週 |
9/26, 9/29 |
Cyclic groups/Cosets & Quotient groups |
第4週 |
10/3, 10/06 |
Normal subgroups/Isomorphism theorems |
第5週 |
10/10, 10/13 |
國慶日放假/Free groups |
第6週 |
10/17, 10/20 |
Group action(I)/Group action(II) |
第7週 |
10/24, 10/27 |
Sylow theorems(I)/Sylow theorems(II) |
第8週 |
10/31, 11/03 |
Simple groups/Semidirect product |
第9週 |
11/7, 11/10 |
Classification of finite groups(I)/Classification of finite groups(II) |
第10週 |
11/14, 11/17 |
Solvable groups/期中考 |
第11週 |
11/21, 11/24 |
Definition of rings & basic properties/Matrix rings |
第12週 |
11/28, 12/01 |
Ideals & quotient rings/Isomorphism theorems & Chinese Remainder theorem |
第13週 |
12/5, 12/08 |
Rings of fractions/Polynomial rings |
第14週 |
12/12, 12/15 |
Euclidean domain & Rings of quadratic algebraic integers/UFD & PID (I) |
第15週 |
12/19, 12/22 |
UFD & PID (II) and Ring of Gauss integers/Gauss lemma |
第16週 |
12/26, 12/29 |
Resultant/Hilbert basis theorem |
第17週 |
1/02, 1/05 |
Grobner basis(I)/Grobner basis(II) |
第18週 |
1/09, 1/12 |
/期末考 |
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