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課程名稱 |
代數一
Algebra(Honor Program)(Ⅰ) |
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開課學期 |
105-1 |
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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 |
MATH5178 |
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課程識別碼 |
221 U6520 |
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班次 |
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學分 |
5 |
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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 |
星期一2,3,4(9:10~12:10)星期四8,9(15:30~17:20) |
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上課地點 |
天數101天數101 |
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備註 |
此課程研究生選修不算學分。
限學士班學生
總人數上限：44人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1051algebra_honor_1 |
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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 |
I. Group theory:
Classical groups, Permutation groups, Cyclic groups, Group homomorphisms, Quotient groups, Group actions, Semidirect products, Simple groups, Composition series, Fundamental theorem of finite abelian groups, Sylow theorems, Classification of finite groups of simple type, Free groups, Free products.
II. Multilinear Algebra:
Group leaving a bilinear form invariant, Tensor product of vector spaces, Tensor algebras, Symmetric algebras, Exterior algebras.
III. Introduction to the representation theory of finite groups:
Linear representations, Character theory, Extensions of groups.
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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. Moreover, 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 |
需修過微積分及線性代數且分數達B以上或線性代數達B+以上。 |
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預期每週課後學習時數 |
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Office Hours |
每週五 16:30~17:20
每週五 15:30~16:20 |
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指定閱讀 |
Knapp, Basic Algebra 2nd Edition.
(作者網頁下載 ：http://www.math.stonybrook.edu/~aknapp/download.html) |
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參考書目 |
Dummit-Foote, Abstract Algebra
N. Jacobson, Basic Algebra I , 2nd edition
Serge Lang, Undergraduate Algebra, 3rd edition
M. Artin, Algebra, 2nd edition
Knapp, Advanced Algebra.
(作者網頁下載 ：http://www.math.stonybrook.edu/~aknapp/download.html) |
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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. |
期中考 |
25% |
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2. |
期末考 |
25% |
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3. |
作業 |
20% |
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4. |
小考4次 |
20% |
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5. |
上台報告和繳交挑戰作業 |
10% |
作業和小考有合理表現的同學才鼓勵參加 |
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週次 |
日期 |
單元主題 |
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第1週 |
9/12,9/15 |
Basic notion & Classical groups;中秋節放假 |
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第2週 |
9/19,9/22 |
Permutation groups and Dihedral groups; 專題報告; groups and internal direct product |
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第3週 |
9/26,9/29 |
Cosets and Quotient groups; 專題報告; Isomorphism theorems |
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第4週 |
10/03,10/06 |
Universal property and direct products & sums; 小考(1); Rings and fields |
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第5週 |
10/10,10/13 |
國慶日放假; Group actions (I) |
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第6週 |
10/17,10/20 |
Group actions (II); 專題報告; Semidirect products |
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第7週 |
10/24,10/27 |
Simple groups and composition series; 小考(2); Modules over PID |
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第8週 |
10/31,11/03 |
Structure of finitely generated abelian groups ; 專題報告; Sylow theorems |
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第9週 |
11/07,11/10 |
Classification of finite groups of simple type; 專題報告; Free groups |
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第10週 |
11/14,11/17 |
期中考; 檢討考卷 + "PID and UFD" |
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第11週 |
11/21,11/24 |
Bilinear forms and Group leaving a bilinear form invariant; 專題報告; Tensor product of vector spaces (I) |
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第12週 |
11/28,12/01 |
Tensor product of vector spaces (II); 專題報告; Tensor algebra |
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第13週 |
12/05,12/08 |
Symmetric and exterior algebra ; 小考(3); Free products |
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第14週 |
12/12,12/15 |
Generalities on linear representations; 專題報告; Character theory (I) |
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第15週 |
12/19,12/22 |
Character theory (II); 專題報告; Applications |
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第16週 |
12/26,12/29 |
Extensions of groups (I); 小考(4); Extensions of groups (II) |
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第17週 |
1/02,1/05 |
元旦補假; 專題報告 |
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第18週 |
1/09, 1/12 |
期末考; 專題報告 |
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