課程名稱 |
代數一 Algebra(Honor Program)(Ⅰ) |
開課學期 |
108-1 |
授課對象 |
理學院 數學系 |
授課教師 |
楊一帆 |
課號 |
MATH5178 |
課程識別碼 |
221 U6520 |
班次 |
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學分 |
5.0 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期三6,7(13:20~15:10)星期五6,7,8(13:20~16:20) |
上課地點 |
天數101天數101 |
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:45人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1081MATH5178_ |
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 |
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 isomorphism theorems; Composition series; Group action and Sylow theorems; The fundamental theorem of finitely generated abelian groups.
2. Ring Theory:
EDs, PIDs and UFDs; Polynomial rings.
3. Field Theory:
Algebraic extensions; Splitting fields and algebraic closures. |
課程目標 |
待補 |
課程要求 |
Linear algebra |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Dummit and Foote: Abstract Algebra, 3rd Edition. |
參考書目 |
待補 |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Homework |
20% |
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2. |
Quizzes |
20% |
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3. |
Midterm |
30% |
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4. |
Final |
30% |
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週次 |
日期 |
單元主題 |
第1週 |
9/11,9/13 |
Preliminaries and some groups; Moon Festival. |
第2週 |
9/18,9/20 |
Dihedral groups, relations and generators; Symmetric groups, homomorphisms and isomorphisms. |
第3週 |
9/25,9/27 |
Subgroups, centralizers, normalizers and kernels, cyclic groups; Generated subgroups, lattice subgroups. [Structure of the multiplicative group of Z/pZ.] |
第4週 |
10/02,10/04 |
Normal subgroups, quotient groups, Lagrange's theorem; Isomorphism theorems, Jordan-Hölder theorem. [Quiz1.] |
第5週 |
10/09,10/11 |
Alternating groups, simpleness of A_n, group actions; Long weekend. |
第6週 |
10/16,10/18 |
Conjugacy classes, class equation; Automorphisms, Sylow theorems. |
第7週 |
10/23,10/25 |
Examples of finite groups, FTFGAG; Primary decompositon theorem for FGAG. [Proof of FTFGAG.] |
第8週 |
10/30,11/01 |
Semidirect products; p-groups, nilpotent groups. [Quiz2.] |
第9週 |
11/06,11/08 |
Solvable groups, free groups; Midterm. |
第10週 |
11/13,11/15 |
Play with algebra. |
第11週 |
11/20,11/22 |
Example of rings; Ring homomorphisms and ideals. [Proof of Wedderburn's little theorem.] |
第12週 |
11/27,11/29 |
Isomorphism theorems, maximal ideal; Prime ideal, ring of fractions, Chinese Remainder Theorem. |
第13週 |
12/04,12/06 |
p-adic integers, ED and PID; UFD and some examples. |
第14週 |
12/11,12/13 |
Gaussian integers, polynomial rings; Eisenstein's criterion, Gröbner basis, Hilbert basis theorem. [Quiz3.] |
第15週 |
12/18,12/20 |
Field extensions, algebraic extensions; Splitting fields. [Cohn's criterion, Perron's criterion.] |
第16週 |
12/25,12/27 |
Algebraic closures, sepersble extensions; Cyclotomic polynomials, Artinian rings. [Quiz4.] |
第17週 |
1/01,1/03 |
New Year’s Day; DVRs and Dedekind domains. |
第18週 |
1/08, 1/10 |
; Final (1/10) |
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