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課程資訊

課程名稱	代數導論二 Introduction to Algebra(Ⅱ)
開課學期	99-2
授課對象	理學院數學系
授課教師	于靖
課號	MATH2106
課程識別碼	201 24220
班次
學分	3
全/半年	半年
必/選修	必帶
上課時間	星期一3,4(10:20~12:10)星期四7,8(14:20~16:20)
上課地點	天數202天數202
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：100人外系人數限制：10人
Ceiba 課程網頁	http://ceiba.ntu.edu.tw/992algebra
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課程概述	Groups : Jordan-Holder theorem, composition series, solvable groups, generators and relations, linear groups. Rings : polynomial rings, matrix rings, rings of algebraic integers, quaternion algebra, Euclidean domains, UFD. Groebner basis, elimination, Hilbert NullstellenSatz. Fields : Galois theory. Modules : Fundamental Theorem of finitely generated abelian groups, Jordan normal forms and rational normal forms.
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.
課程要求	Course prerequisite： Only students who have taken “Algebra I ” and have past “Algebra I ” are eligible to take this course.
預期每週課後學習時數
Office Hours
指定閱讀	J. Rotman: First Course in Abstract Algebra, Chapter 7, 3rd ed. Prentice Hall 2005.
參考書目	Textbook J. Rotman: First Course in Abstract Algebra, 3rd ed. Prentice Hall 2005. M. Artin, Algebra 2nd edition, Prentice Hall 2010. References D. Cox, J. Little, D. o'shea, Ideals, Varieties, and Algorithms, 3rd edition, 2007, UTM, Springer. N. Jacobson, Basic algebra I ,
評量方式 (僅供參考)

課程進度

週次	日期	單元主題
第1週	2/21,2/24	Solving systems of polynomial equations. Review of linear algebra, systems of linear equations. Division algorithms, Monomial orders, Non-linear Gaussian eliminations. Maximal ideals.
第2週	2/28,3/03	similarity of matrices, canonical forms.
第3週	3/07,3/10	Noetheriaian rings, Hilbert basis theorem, Hilbert Nullstellensatz.
第4週	3/14,3/17	Quadratic number fields, ring of integers, unique factorization.
第5週	3/21,3/24	Groeber basis, Buchberger's algorithm, elimination theory
第6週	3/28,3/31	Correspondence between algebraic sets and radical ideals
第7週	4/04,4/07	Applications of Groebner basis
第8週	4/11,4/14	Modules, examples and constructions
第9週	4/18,4/21	Row and column operations. Diagonalization of matrices
第10週	4/25,4/28	Generators and relations, submodules of finitely generated modules
第11週	5/02,5/05	Rational canonical forms, structure theorem
第12週	5/09,5/12	Galois theory revisted
第13週	5/16,5/19	Galois correspondences
第14週	5/23,5/26	Cubic and qartic equations
第15週	5/30,6/02	Cyclotomic fields
第16週	6/06,6/09	Kummer extensions, quintic equations
第17週	6/13,6/16	Applications of field theory