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課程名稱 |
代數一
Algebra(Honor Program)(Ⅰ) |
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開課學期 |
111-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
莊武諺 |
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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 |
5.0 |
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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101天數101 |
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備註 |
初選不開放。此課程研究生選修不算學分。
限學士班學生 且 限學士班二年級以上
總人數上限：30人 |
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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 |
groups, rings, fields, Galois theory, modules, group representations.
class recording: (usually one to two weeks behind class schedule, due to video editing.)
https://www.youtube.com/watch?v=oDurTPRF_M4&list=PLQqeHUV7RZ2OmZ2AjTbTAu-n04LfH_Mjt&ab_channel=ZenTsai |
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課程目標 |
We will introduce basic concepts in algebra, including groups, rings, and fields, and cover more advanced topics. We hope to equip students with a solid foundation in algebra. |
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課程要求 |
此課程之修課要求為需修過數學系微積分及線性代數、或修過線性代數且成績達B以上。選修授權碼將於開學後給予（請與授課教師email聯繫）。 |
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預期每週課後學習時數 |
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Office Hours |
每週四 13:30~15:30 備註： Office hour at Astro-Math RM403. |
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指定閱讀 |
Lang, Algebra, 3rd edition. (available online through NTU library)
https://link.springer.com/book/10.1007/978-1-4613-0041-0 |
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參考書目 |
Jacobson, Basic Algebra I & II, 2nd edition.
Dummit and Foote, Abstract Algebra, 3rd edition.
Knapp, Advanced Algebra, digital 2nd edition. (available on the author's website)
Other references will be supplemented along the way. |
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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. |
homework |
30% |
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2. |
midterm |
35% |
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3. |
final |
35% |
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週次 |
日期 |
單元主題 |
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第1週 |
09/07, 09/09 |
09/07: basic notion, symmetry groups.
09/09: no class. |
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第2週 |
09/14, 09/16 |
09/14: cosets, normal subgroups, quotient groups, isomorphism theorem. HW1 due.
09/16: isomorphism theorem, tower of subgroups. |
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第3週 |
09/21, 09/23 |
09/21: butterfly lemma, Jordan-Holder theorem. HW2 due.
09/23: Jordan-Holder theorem, cyclic groups. HW3 due. |
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第4週 |
09/28, 09/30 |
09/28: no class.
09/30: group action, simpleness of A_n. HW4 due. |
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第5週 |
10/05, 10/07 |
10/05: simpleness of A_n, Sylow theorems.
10/07: Sylow theorems, HW5 due. |
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第6週 |
10/12, 10/14 |
10/12: semidirect products. HW6 due.
10/14: structure theorem of finite abelian groups. |
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第7週 |
10/19, 10/21 |
10/19: ring isomorphism theorem. HW7 due.
10/21: Chinese remainder, localization. |
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第8週 |
10/26, 10/28 |
10/26: ED, PID, UFD. HW8 due.
10/28: midterm. |
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第9週 |
11/02, 11/04 |
11/02: Gauss lemma, criteria for irreducibility. HW9 due.
11/04: algebraic extensions. |
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第10週 |
11/09, 11/11 |
11/09: algebraic closures. HW10 due.
11/11: algebraic closures, normal extensions. |
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第11週 |
11/16, 11/18 |
11/16: separable extensions.
11/18: primitive element theorem, finite fields. HW11 due. |
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第12週 |
11/23, 11/25 |
11/23: Galois theory. HW12 due.
11/25: Galois theory. |
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第13週 |
11/30, 12/02 |
11/30: examples and applications of Galois theory, roots of unity, cyclotomic extensions. HW13 due.
12/02: no class. (台大運動會) |
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第14週 |
12/07, 12/09 |
12/07: norms and traces. HW14 due.
12/09: purely inseparable extensions, cyclic extensions. |
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第15週 |
12/14, 12/16 |
12/14: solvable extensions, solvable by radicals. HW15 due.
12/16: final. |
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