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課程名稱 |
代數導論二
Introduction to Algebra (Ⅱ) |
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開課學期 |
101-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
王姿月 |
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課號 |
MATH2106 |
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課程識別碼 |
201 24220 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) |
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上課地點 |
天數204天數204 |
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備註 |
教學改善計畫課程有教學助理實施小班輔導。時段:四8
總人數上限：80人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1012abstractalgebraW |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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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 |
Fields : Galois theory
Groups : p-groups and the Sylow theorems, Jordan-Holder theorem, composition series, solvable groups.
Rings : unique factorization domains, principal ideal domains, rings of algebraic integers
Modules : modules over PID, fundamental theorem of finitely generated abelian groups |
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課程目標 |
*Introducing algebraic structures, algebra as a basic tool in mathematics. Using the language of Algebra. Method of Algebra. Applications of Algebra. From Linear Algebra to Nonlinear Algebra. Classification of algebraic structures. |
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課程要求 |
Course prerequisite：
linear algebra, Chapter 2,3,4,6 of the text book which includes groups, rings, polynomial rings, fields
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預期每週課後學習時數 |
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Office Hours |
每週四 14:00~15:00 |
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指定閱讀 |
教科書: W. K. Nicholson, Introduction to Abstract Algebra, 3rd edition, John Wiley & Sons, 2007 |
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參考書目 |
T. W. Judson, Abstract Algebra Theory and Applications
(can be obtained via http://abstract.pugetsound.edu/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. |
Homework |
10% |
Homework will be assigned every week. There will be 10-15 problems for each homework. 5 problems will be graded carefully and each of them will be 15 points. The remaining problems will be given 25 points in total if they are done with work shown. |
2. |
Project |
10% |
There will be two projects (due during miterm exam and final exam). |
3. |
Quiz |
20% |
Every two weeks |
4. |
Midterm Exam |
30% |
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5. |
Final Exam |
30% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/18,2/21 |
review of field theory I, examples of Galois groups, separable extension |
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第2週 |
2/25,2/28 |
primitive element theorem, basic properties of separable polynomials and extensions, review of splitting fields |
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第3週 |
3/04,3/07 |
separability v.s. Galois group, basic definitions and examples prepared for the main theorem of Galois theory |
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第4週 |
3/11,3/14 |
the main theorem of Galois theory |
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第5週 |
3/18,3/21 |
Equivalent conditions for finite Galois extension, character, Dedekind's Lemma, Dedekind-Artin theorem |
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第6週 |
3/25,3/28 |
radical extension, Galois criterion, unsolvable polynomial of degree 5 |
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第7週 |
4/01,4/04 |
Cyclotomic extension, kummer extension, solvable groups |
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第8週 |
4/08,4/11 |
solvable groups, symmetric polynomials, general non-solvable polynomials |
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第9週 |
4/15,4/18 |
cyclotomic extension, review of Galois theory, midterm exam |
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第10週 |
4/22,4/25 |
correspondence theorem of groups, isomorphism theorems of groups, a brief introduction on classification of finite simple groups |
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第11週 |
4/29,5/02 |
Cauchy's theorem, p-groups, class equation from conjugacy class |
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第12週 |
5/06,5/09 |
group actions, application of the Sylow theorems |
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第13週 |
5/13,5/16 |
the Sylow theorems, classification of groups up to order 15 |
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第14週 |
5/20,5/23 |
unique factorization domain(UFD), ascending chain condition on principal ideals(ACCP) |
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第15週 |
5/27,5/30 |
Gauss Lemma, R[x] is UFD if R is UFD, PID, Euclidean domains |
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第16週 |
6/03,6/06 |
quadratic fields, rings of integers of quadratic fields |
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第17週 |
6/10,6/13 |
ideal class group of imaginary quadratic fields, some Diophantine equations, class equation of the icosahedral group |
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