|
課程名稱 |
代數二
Algebra(Honor Program)(Ⅱ) |
|
開課學期 |
105-2 |
|
授課對象 |
理學院 數學系 |
|
授課教師 |
林惠雯 |
|
課號 |
MATH5179 |
|
課程識別碼 |
221 U6530 |
|
班次 |
|
|
學分 |
5.0 |
|
全/半年 |
半年 |
|
必/選修 |
選修 |
|
上課時間 |
星期一2,3,4(9:10~12:10)星期四8,9(15:30~17:20) |
|
上課地點 |
天數101天數101 |
|
備註 |
總人數上限：40人 |
|
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1052Algebra_Honor_2 |
|
課程簡介影片 |
|
|
核心能力關聯 |
本課程尚未建立核心能力關連 |
|
課程大綱
|
|
為確保您我的權利,請尊重智慧財產權及不得非法影印
|
|
課程概述 |
I. Fields and Galois theory
Field extensions, the Fundamental theorem of Galois theory, Finite fields, Solution by radicals, Abelian extension, Kummer extension, Calculations of Galois groups, Transcendental extensions. (挑戰作業題組: Resultants, Infinite Galois theory )
II. Introduction to Commutative Algebra
ED+PID+UFD, Primary decomposition, The equivalence of geometry and algebra, Grobner basis, Going-up and Going-down theorems, Local rings, Artin rings, Dedekind domains. (挑戰作業題組: Scheme theory)
III. Introduction to Homological Algebra
Projective & Injective & Flat modules, Ext & Tor functors, Koszul complex, Derived & Homotopy categories, Derived functor. |
|
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. |
|
課程要求 |
需修過代數一且分數達C+以上。 |
|
預期每週課後學習時數 |
|
|
Office Hours |
|
|
指定閱讀 |
Knapp, Basic Algebra 2nd Edition.
(作者網頁下載 ：http://www.math.stonybrook.edu/~aknapp/download.html) |
|
參考書目 |
Dummit-Foote, Abstract Algebra
N. Jacobson, Basic Algebra I & II
Serge Lang, Undergraduate Algebra, 3rd edition
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) |
|
評量方式
(僅供參考) |
|
No. |
項目 |
百分比 |
說明 |
|
1. |
期中考 |
25% |
|
2. |
期末考 |
25% |
|
3. |
平常成績 |
24% |
作業表現 ; 由助教們決定 |
4. |
小考 2 次 |
16% |
|
5. |
上台報告和繳交挑戰作業 |
10% |
|
|
|
週次 |
日期 |
單元主題 |
|
第1週 |
2/20,2/23 |
Algebraic extensions; Construction of field extensions |
|
第2週 |
2/27,3/02 |
228假期; Finite fields |
|
第3週 |
3/06,3/09 |
Algebraic closures; 專題報告: Geometric constructions by Straightedge and Compass (王敏齊); Separable extensions |
|
第4週 |
3/13,3/16 |
Normal extensions; 專題報告: Primitive theorem (朱雅琪); Fundamental theoreom |
|
第5週 |
3/20,3/23 |
Abelian extensions; 小考; Kummer extensions |
|
第6週 |
3/27,3/30 |
Solution by radicals; 專題報告: Construction of regular polygons (莊秉勳); Ruffini-Abel theorem; 專題報告: Purely inseparable extensions (江誠敏) |
|
第7週 |
4/03,4/06 |
春假; Calculation of Galois groups |
|
第8週 |
4/10,4/13 |
Transcendental extensions; 專題報告: \pi is transcendental (廖泰甯); Hilbert theorem 90&Norma base |
|
第9週 |
4/17,4/20 |
期中考; ED + PID + UFD |
|
第10週 |
4/24,4/27 |
Primary decomposition; 專題報告: Gauss lemma (茅耀文); The equivalence of geometry and algebra |
|
第11週 |
5/01,5/04 |
Grobner basis (I); 專題報告: Hilbert basis theorem (李堂愷); Grobner basis (II) |
|
第12週 |
5/08,5/11 |
Local rings; 專題報告: Localization (黃凱祺); Krull dimension |
|
第13週 |
5/15,5/18 |
Artin rings; 專題報告: Noether normalization (陳冠文); Dedekind domains |
|
第14週 |
5/22,5/25 |
Projective , Injective & flat modules; 小考; Homology functors |
|
第15週 |
5/29,6/01 |
端午節假期; Ext & Tor |
|
第16週 |
6/05,6/08 |
Koszul complexes; 專題報告: Mayer-Vietoris sequence (周子揚); Derived category(I) |
|
第17週 |
6/12,6/15 |
Derived category(II); 專題報告: Mapping cylinder (王毓平); Homotopy category(專題報告(望開翔)); Derived functor |
|
第18週 |
6/19 |
期末考 |
|