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課程名稱 |
代數二
Algebra(Honor Program)(Ⅱ) |
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開課學期 |
109-2 |
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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 |
MATH5179 |
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課程識別碼 |
221 U6530 |
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班次 |
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學分 |
5.0 |
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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 |
星期三6,7(13:20~15:10)星期五6,7,8(13:20~16:20) |
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上課地點 |
天數102天數102 |
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備註 |
此課程研究生選修不算學分。
限本系所學生(含輔系、雙修生)
總人數上限：45人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1092MATH5179_algeb2 |
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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. Module theory
Definitions & Properties & Examples ; Free modules ; Modules over a PID ; Structure theorem for modules over a PID ; Tensor product ; Modules of fractions ; Noetherian modules ; Primary decompositions ; Nakayama's & Artin-Rees lemmas ; Hilbert polynomial ; Indecomposable modules.
II. Homological algebra
Projective, Injective & Flat modules ; Homology functors ; Ext & Tor ; Koszul complex ; Extensions of abelian groups ; 1st and 2nd group cohomology ; Applications of group cohomology.
III. Representation theory
Semisimple modules & rings ; Simple rings ; Representations & semisimplicity ; Characters ; Orthogonality ; Induced representations ; Regular character ; Brauer's theorem ; GL_2 over a finite field. |
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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 |
需修過代數一且分數達B以上。 |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Serge Lang, Algebra, 3rd edition |
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參考書目 |
Dummit-Foote, Abstract Algebra
N. Jacobson, Basic Algebra I & II
M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra
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 |
35% |
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2. |
平常成績 |
30% |
作業表現 ; 由助教們決定 |
3. |
上台報告 |
0% |
爭取A+的機會 |
4. |
期末考 |
35% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/24,2/26 |
Definitions & Properties & Examples / Free modules |
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第2週 |
3/03,3/05 |
Modules over a PID / Structure theorem and applications |
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第3週 |
3/10,3/12 |
Tensor product / Modules of fractions |
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第4週 |
3/17,3/19 |
Noetherian modules / Primary decompositions |
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第5週 |
3/24,3/26 |
Nakayama's lemma & Artin-Rees lemma / The Hilbert polynomial |
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第6週 |
3/31,4/02 |
Indecomposable modules / 補假 |
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第7週 |
4/07,4/09 |
Projective, Injective & Flat modules / Homology functors |
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第8週 |
4/14,4/16 |
Ext / Tor |
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第9週 |
4/21,4/23 |
Koszul complex / Extensions of abelian groups |
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第10週 |
4/28,4/30 |
習題課 / 期中考 (考到 Ext and Tor) |
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第11週 |
5/05,5/07 |
1st and 2nd group cohomology / Applications |
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第12週 |
5/12,5/14 |
Semisimple modules / Semisimple rings |
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第13週 |
5/19,5/21 |
Simple rings / Representations & semisimplicity |
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第14週 |
5/26,5/28 |
Characters / Divisibility & Burnside's theorem |
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第15週 |
6/02,6/04 |
Examples / Induced representations(I) |
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第16週 |
6/09,6/11 |
Induced representations(II) / Brauer's theorem (I) |
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第17週 |
6/16,6/18 |
Brauer's theorem (II) / GL_2 over a finite field |
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第18週 |
6/25 |
期末考 |
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